ΒιογραφικοΣημειωμα...
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Βιογραφικο Σημειωμα Α. Τσολομυτης Προσωπικά στοιχεία Αντώνης Τσολομύτης Πανεπιστήμιο Αιγαίου Τμήμα Μαθηματικών 832 00 Καρλόβασι Σάμος Τηλ. : 30 22730 82123 Em@il: [email protected] Web: http://myria.math.aegean.gr/~atsol Ημερομηνία Γεννησης: 25/6/1967 στον Πειραιά. Παντρεμένος με τρία παιδιά. Υπηκοότητα Ελληνική. Υπηρεσία στον Ελληνικό Στρατό: Ιούνιος 1996–Ιανουάριος 1998. Εκπαίδευση Ιούνιος 1989 Πτυχίο στα Μαθηματικά από το Πανεπιστήμιο Αθηνών (Βαθμός: Άριστα, 9,18). Ιούνιος 1996 Διδακτορική Διατριβή (Ph.D.) από το Ohio State University. Τίτλος: Symmetrizations and convolutions of Convex Bodies υπό την επίβλεψη του V. D. Milman. Διακρίσεις 1995–1996 Presidential Fellow στο Τμήμα Μαθηματικών του Ohio State University, Columbus, Ohio, usa. 11 Δεκεμβρίου 2009 «Βραβείο Αριστείας στη Διδασκαλία» της Σχολής Θετικών Επι- στημών του Πανεπιστημίου Αιγαίου.
Transcript of ΒιογραφικοΣημειωμα...
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.
83200
. : 30 22730 82123Em@il: [email protected]: http://myria.math.aegean.gr/~atsol
: 25/6/1967 . .
. : 1996 1998.
1989 (:, 9,18).
1996 (Ph.D.) Ohio State University. :Symmetrizations and convolutions of Convex Bodies V. D. Milman.
19951996 Presidential Fellow Ohio State University,Columbus, Ohio, usa.
11 2009 - .
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. G 2/16
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2. Functional analysis, an introduction, with Y. Eidelman and V.D. Mil-man. Graduate Studies in Mathematics, no 66, AMS, 2004 .
3. , . - ar. 7, LeaderBooks, 2004.
4. Digital Typography using LATEX. With A. Syropoulos and N. Sofro-niou. Springer Profesional Computing, Springer-Verlag, New YorkOct. 2002.
(subj. class. AMS2010 52A21)
1. Geometry of random sections of isotropic convex bodies, withA. Giannopoulos and L. Hioni (submitted).
2. Remarks on the Rogers-Shephard inequality, with A. Giannopou-los and E. Markessinis, Proc. Amer. Math. Soc. 144 (2016), no. 2,763773.
3. Asymptotic shape of the convex hull of isotropic log-concaverandom vectors, with A. Giannopoulos and L. Hioni, Adv. in Appl.Math. 75 (2016) 116143.
4. Geometry of the centroid bodies of an isotropic log-concavemeasure, with A. Giannopoulos, P. Stavrakakis and B-H. Vritsiou,Trans. AMS 367 (2015), 45694593.
5. Quermaintegrals and asymptotic shape of random polytopes inan isotropic convex body, with N. Dafnis and A. Giannopoulos,Michigan Mathematical Journal, vol. 62, Issue 1, (2013) 5979.
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. G 3/16
6. Asymptotic shape of random a polytope in a convex body, withN. Dafnis and A. Giannopoulos, J. Funct. Anal. (2009),doi:10.1016/j.jfa.2009.06.27.
7. Random points in isotropic unconditional convex bodies, withA. Giannopoulos and M. Hatrtzoulaki, JLMS (2005), 72: 779798.
8. Asymptotic formulas for the diameter of sections of symmetricconvex bodies, with A. Giannopoulos and V. D. Milman, J. Funct.Anal. 223 (2005) 86108.
9. Volume radius of a random polytope in a convex body, withA. Giannopoulos. Math. Proc. Cambridge Philos. Soc. 134, 2003,no. 1, 1321.
10. Johns theorem for an arbitrary pair of convex bodies, with A. Gian-nopoulos and I. Perissinaki, Geometri Dedicata 84, (2001), 6379.
11. A note on the -limiting convolution body, Covnex Geometry,msri Publications, Volume 34, 1998.
12. Convolution bodies and their limiting behavior, Duke Math. Journal87 (1997), no. 1, 181203.
13. On the convolution body of two convex bodies, C.R. Seances Acad.Sci. Ser. I 322, 1, (1996) 6367.
14. Quantitative Steiner/Schwarz-type symmetrizations, GeometriDedicata 60: 187206, 1996.
(subj. class. AMS2010 68U15)
15. A direct TeX-to-Braille transcribing method, with A. Papasalouros(submitted)
16. A direct TeX-to-Braille transcribing method, with A. Papasalouros,ASSETS 15: 17th International ACM SIGACCESS Conference onComputers & Accessibility, October 2628, 2015, Lisbon, Portugal,ACM 978-1-4503-3400-6/15/10.http://dx.doi.org/10.1145/2700648.2811376(to appear)
17. The Kerkis Font Family, TUGboat, Vol. 23, No. 3/4, 296301, 2002,(invited paper).
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(subj. class. AMS2010 68U15)
1. TrueType TEX LATEX, , 11/12, 16, 2004.
2. , . . , 8, 2527, 2002.
3. LATEX2e, - . . , 3, 5768, 2000.
4. LATEX TrueType, . . , 2, 1722, 1999.
( )
1. . - . 3 4 .
2. ( , ) (2013).
3. IV: Green,Stokes Gauss. - . (2012)
4. ( ) (2009).
5. . (2003).
-
( )
-
http://myria.math.aegean.gr/
-
. . . -
.
. . Jensen
.. H -
.. H Young .. H Hlder .. H Hlder .. H Minkowski .. H Minkowski
.
. , . . . .
-
.
.. . 1
. Hausdorff . Hausdorff . Blaschke
- - . Prekopa-Leindler . Brunn-Minkowski . . Brunn . Borel
. Steiner
. Gross . ( ) . Blaschke-Santal
.
. -
.
.
.. .. .. .. -
-
..
. .. Binet Legendre
. &
. . . john
. Brascamp-Lieb Barthe . Brascamp-Lieb Barthe
. Brascamp-Lieb Barthe
.
. .
. .
.
G
. . Laplace
-
.
.
() (1, 2, , ) (1, 2, , ) ,
2(, ) = (1 1)2 + (2 2)2 + + ( )2.
, () 2(0, ) 2. ,
2 = 21 + 22 + + 2 .
2(, ) = 2 , . 2 2(, ).
() > 0 , (, ). ,
(, ) = { 2 < }.
.. ( ). - . 0 , > 0 (0, ) . int.
. , () 3. :
.. ( ). - . relint()
-
.
.. ( ). - > 0
(, ) {} .
. = .
() 2 0. , = , .. = 1/ (, 1/) . 2 < 1/ 0.
= . , = .
... (0, ) - 2(0, ).
2(0, ) = { 2 }.
, 2(0, ) () (0, ) 2 0. 2 = 1, 2, ,
| | 2 0.
, 2 2 2. 2 2 . , 2 < 1 (0, ) 2(0, ). , 2 = (1 1/), , (1 1/) < , (0, ) . (0, ), 2(0, ).
2(0, ) (0, ) , 2(0, ) = 2(0, ).
-
... 2(0, 1) 2.
bd(2) 2 { 2 = 1}.
... { 2 = 1} () 1.
...
(i) + { + }.
(ii) { }.
2(0, ) = 2 2(, ) = + 2.
.. ( ). > 0 2 . 2.
... . .
: 2. 2 = 2.
.. ( ). .
... , . , =
(, ) = inf{ 2 , } = 2 > 0.
:
= inf{ 2 , }.
, 2 . , ,
-
. . 2 . ,
2 2 + 2.
( ) . > 0 ,
2(0, ). 2(0, )
, . , , , . , . . , , .
= lim
= 2.
, , . , .
.. (). (, ) (, ) 0 > 0 > 0 (, 0) < ((), (0)) < .
.. ( ). > 0 > 0, [, ] = 1, 2, ,
=1( ) <
=1
|() ()| < .
.. ( Lipschitz). Lipschitz , > 0
|() ()| | |,
, . Lipschitz .
Lipschitz .
-
... (, ) (, ) . . , .
.. bd(2) = 1.
..
() ={{
sin 1 (0, 1],
0 = 0,
. (: - [(/2)1(4(( + ) + 1))1, (/2)1(4( + ))1] = 0, 1, , (2)1.)
.
.. ( ).
, ,
(i) , 0 .
(ii) , = 0 = 0.
(iii) , = , .
(iv) , 11 + 22 = 1, 1 + 2, 2, 1, 2 .
, =
=1
,(.)
. ( )
-
(.). , ( ). , = 22 . :
.. ( Cauchy-Schwartz). , -
|, | 22.
: ,
0 , = 22 + 222 2, ,
.
4, 2 42222 0,
. ,
.
.. (). :
(i) 0. 0 , 0 = 0 = 0 = 0.
(ii) : = || , , .
(iii) : + + , , .
2 . 2 (i) (ii) . H (iii) .. .
, :
-
.. ( ). , -
.
:
= ( ) + +
. ,
= ( ) + +
= .
.
+ .
(, ) = , . (, ) . . , . .
.. ( ). - .
: , 2 0 2 0, , , . |, , | 0. Cauchy-Scwhartz ( ..)
, , = , , + , , = , + , 2 2 + 22 0,
2 0, 2 0 2
-
.. ( ). ( ) .
: , 2 2. ( ..),
2 2 2 0. (
) .
... , > 0 ,
2 2.
: 1, 2, , . 1, 2, , ,
=
=1
=1
=
=1
|| = (||)
=1, ()
=1
(
=1
||2)
12
(
=1
2)
12
(.)
(.) Cauchy-Schwartz
(||)=1 ()=1. = (=1
2)1/2
, (.) 2.
() = . , , () ().
|() ()| = 2 0.
1 = { 2 = 1} , . , , 1. = min1 (). , 0 1 (0) = ,
-
0 = . = 0, 0 = 0 0 = 0 1, . > 0. {0} /2 1 (/2) /2 2. = 0 .
... - .
Proj
. 1, , ( ) 1, , , +1 , .
=1
1 , , = , = 1, , .
Proj() =
=1
.
Proj ,
Proj()2 = (
=1
||2)1/2
(
=1
||2)1/2
2,
Proj .
.
, ,
[, ] = {(1 ) + [0, 1]}.
.. , . , [0, 1] = + ( ). = (1 ) + .
-
Y
.:
.: . .
.. ( ). , . , ,
[, ] = {(1 ) + [0, 1]}
.. ( ). , (0, 1) (1 ) + int().
0, 0
0, = { 0, = 0}
= { , = 0, }
-
0, = { , 0, } +0, = {
, 0, }.
+0, 0. 0, (
0)
, = { , = }
. ,
, = { , } +, = { , }.
, 0, , ,
{ , = 1},
.
.
,, = { , } = +, ,,
, . ,
+, ,,
,, +,,.
.. ( affine-). 1, , affine- 2 1, 3 1, , 1 .
... 1, , + 1, + 2 + 1 2, , +2 1 , dim = .
.. ( ). -. . .
-
() = ( ).
, 1. = (1, , ) . = (1, , ) = (1, , ) ,
, =
12
,
12
= (1, 2, , )
12
=
. ,
(), = () = () = () = , ().
.
... 1, , 1, ,
= {
=1
, 2
2 1} .
1, , -.
-
I
-
.
.. (K ). -
((1 ) + ) (1 )() + ().
[0, 1] , .
.. ( ).
((1 ) + ) < (1 )() + ()
(0, 1) , .
...
((1 ) + ) = (1 )() + (),
(0, 1), = .
... . , [0, 1] (1 ) + . , = 0 = 1 . {0, 1} () = (1)+ > 0. = 1. () (1) = 0
, = () (1 ) +
(1)+ (1 ) + ,
, [0, 1].
-
, > 1, . () > (1) = 0 ( , , ) > .
..
(i)1 + 1 + 2
(ii) (iii)1
1 + + 2() (log )2
. .. , + . .. , sup () .
.. , ( ) . ; .. , , , < < ,
() ()
() ()
() ()
.
.. . , int() , . , int(). (: < < . , < .. < < < < < < . () () < 0 () () 0.) .. [0, +) (0) = 0 () = ()/ {0}.
.. .. .
-
.. () , () +1() , . . () () , . .. , . , + , +. . .
(: 1: .. < < < < < < < < < < +
() +() () ()
() +(), (*)
.
2: () () =(( 1/) ())/(1/) . .. . + . .
3: (*) < () +() (). + .) .. . :
(i) H .
(ii) H .
(: , .. .) .. , . 0. ..
() () + ( )
-
. , :
(i)
() 0 (0) = 0. (0, +) (, 0) (0, 1) 0 = (1)+ ,
sup(,0)
()
inf(0,+)
()
.
()
sup(,0)
()
inf(0,+)
()
,
0.
() , () = ( + ) () (0) = 0.
(ii) , () = () () = sup () . .
.
.
... , 1, 2, , 1 + 2 + + = 1 [0, 1] = 1, , . 11 + 22 + +
(11 + 22 + + ) 1(1) + 2(2) + + ().
: Jensen . = 2: 1 + 2 = 1, 1 = 1 2,
(11 + 22) = ((1 2)1 + 22) (1 2)(1) + 2(2) 1(1) + 2(2)
-
. + 1 . 1, 2, , +1 , 1, , +1 0
+1=1 =
1. +1 = 1 1 = = = 0. ,
(+1+1) +1(+1). (+1) (+1), .
0 +1 < 1 :
(11 + 22 + + +1+1)
= ((1 +1) (1
1 +11 + +
1 +1
) + +1+1)
(1 +1) (1
1 +11 + +
1 +1
) + +1(+1),
. : /(1 +1) 0,
=1
(
1 +1) =
1 +11 +1
= 1.
(11 + 22 + + +1+1)
(1 +1) (1
1 +1(1) + +
1 +1
()) + +1(+1)
= 1(1) + 2(2) + + +1(+1).
Jensen.
... (0, 1) = 1, ,
=1 = 1
(11 + + ) = 1(1) + + ()
1, , 1 = 2 = = .
-
:
1 1
1 1
=1
() = (
=1
)
= (11 + (1 1) 1
1 1
)
< 1(1) + (1 1) (1
1 1
) ,
=1
(),
. 1 = 1
11. , = 1, ,
=
1. ,
(1 1)1 = 1
= 2
11 + 22
= (1 2)2 11 + 22= 2 11.
1 = 2. 1. Jensen ,
( Riemann ) ( ) ( ) 1. .
.. ( Jensen ). [, ] Riemann = 1
-
(
() )
( )() .
, (, )
(
)
.
: Riemann ( ..),
(
() ) = lim
()
= 1,
lim
(())
( )() .
0 = .. , + () 0 + = (0). (()) () + . () = 1 0
( ) 0 + = (0) = ( ) ,
-
.. H -
... 1, 2, , 0, 1, 2, , 0,
=1 = 1. :
(.) 1122 11 + 22 + + .
> 0 = 1, , .
:
(.) 12 1 + 2 + +
,
.
: (.) (.) 1 = = = 1/ 0.
=1 = 1, :
1/1 1/ 1
1 + +1
,12
1 + 2 + +
.
, 1/ > 0. (.)
. () = . ... 1, 2, , 0. 1 = ln 1, , = ln Jensen , 1, , 1, ,
11++ 11 + + .
1 ln 1++ ln 1ln 1 + + ln ,
,ln(1)
1++ln() 11 + 22 + + ,
1122 11 + 22 + + .
-
, , ( ..) .. , .. H Young
... , 0 , , > 1 1 + 1 = 1,
+
.
= .
: = 0 = 0 . ,
1 =1
, 2 =1
, 1 = , 2 = ,
... 11 22 11 + 22
()1 ()
1
1
+1
,
. ... .. H Hlder
... 1,1, 2,2, , , 0, , > 1 1+1 =1
11 + 22 + + (1 + + )
1 (1 + + )
1 .
= 0 = 1, , =
= 1, 2, , .
: . , 1 + +
1 + + .
-
Young
=
(1 + + )1/
=
(1 + + )1/
,
(=1
)
1 (
=1
)
1
(=1
)
+
(=1
)
,(.)
,
=1
(1 + +
)
1 (1 + +
)
1
1
(1 + + )
=1
+1
(1 + + )
=1
= 1.
,
=1
(1 + + )
1 (1 + + )
1 .
= = 1, 2, ,
. = 0 = 1, , .
, , = 0 (.) = 1, 2, , ( ). (.) = 1, 2, , . Young
1 + +
=
1 + +
= 1, , . = = 1, ,
=1 + + 1 + +
,
.
-
.. H Hlder
- Lebesgue. [0, ), , = 0 .
... , - , > 1 1 + 1 = 1. ,
(
)
1
()
1
.
= 0 = .
: = 0 = 0 -
( ). = 0.
0
0. Young :
( )
1 (
)1
+
,(.)
,
( )
1 (
)1
1
+
1
= 1,
. ,
= 0 = 0 . = 0.
0 - (.) ( ) - Young = , = (
)1//( )1/.
-
.. H Minkowski
... 1, 1, , , 0 1,
((1 + 1) + + ( + ))1 (1 + + )
1 + (1 + + )
1 .
= 1 = 0 = 1, , = = 1, , .
: = 1 ( ). > 1. > 1 1 +1 = 1, = /( 1).,
=1
( + ) =
=1
( + )( + )1
=
=1
( + )1 +
=1
( + )1
(
=1
)
1
(
=1
( + )(1))
1
+ (
=1
)
1
(
=1
( + )(1))
1
=
(
=1
)
1
+ (
=1
)
1
(
=1
( + ))
1
.
,
(
=1
( + ))1 1
(
=1
)
1
+ (
=1
)
1
.
1 1
= 1, .
- ... .. H Minkowski
Hlder -,
-
Lebesgue.
... , - 1,
(( + ))
1
()
1
+ ()
1
.
= 1 = 0 = .
: ( + ) = 0 = = 0
( ). ( + ) 0.
= 1 ( ). > 1 :
( + ) =
( + )( + )1
= ( + )1 +
( + )1
()
1
(( + )(1))
1
+ ()
1
(( + )(1))
1
=
()
1
+ ()
1
(( + ))
1
.
,
(( + ))
1 1
()
1
+ ()
1
.
1 1
= 1, .
.
-
.. Young , .
.. 0 < <
(||)
1
(||)
1
,
, ( ) ( ) 1.
.. () = (0, ) Jensen
(||)
1
||,
, ( ) ( ) 1.
.
Minkowski 1
= (
=1
||)1/
( ) . , . , ( (, ) = ), , . ,
| | = (
=1
| |)1/
.
-
= max1 ||.
Minkowski 1 [, ]
= ([,]
||)1/
. . [, ] . , . , () = [0, 1] () = 1 [1, 2] [0, 2] Cauchy , >
= 1
0( )
1
0 =
1 + 1
0
. [0, 2] : [0, 2] 0. () = 0 [0, 1) () = 1 [1, 2]. 1 [0, 2]. 0.
+ 0,
= 0. [1, 2] [0, 1 1/] . 11/0 | |
20 | | = 0, =
[0, 1 1/] . = [1, 2]. = [0, 2] .
([, ], ) [, ].
.. = max1 || , ,
-
=1
||
1/
. .. 1 . .. 0 < < 1 . = { 1} 0 < < 1 . ( 3 = 0,6 = 0,8 .) .. 1 2. -, 1 < <
1
1 .
-
. ,
.. ( ). . , , int() , .
.. ( ). . , 0.
... 0 : 0 int().
: > 0 (, ) . , (, ) .
12
(, ) +12
(, ) .
(0, ) 12
(, ) +12
(, ).
(0, ), 2 < , =12(+)+ 1
2(),
+ (, ) (, ). ( + )2 = 2 < () ( )2 = 2 < .
:
...
= { 1}
, .
-
: ( 0 ) ( = ). , , [0, 1]
(1 ) + (1 ) + (1 ) + = 1,
(1 ) + . , ( ) (, 2).
( ..): , . 1, 1
... , > 0 2 2 . , 1 2 1/. 2(0, 1/) . , (0, 1/) 2 < 1/. 2 1 . (0, 1/) .
... ,
= { 1},
() (, ).
... , . - . .
.. ( o ). =max1 || ()=1 . , { 1}, 2 -, . [1, 1]. 4. = 2 [1, 1] [1, 1] 2.
-
,
.: 3 3.
.. ( 1). 1 (1, 2, , )
= (
=1
||)1/
,
( - ..).
{ 1}
, ; (, ). . . 31 , (1, 2, 3) 3 |1|+|2|+|3| 1, ( crosspolytope).
.: 31 (3, 1).
1 < < 1 .
-
, 1 || 1 ., < || ||
=1
||
=1
|| 1.
1. , /
c
.: 21 23/2 22 23 2.
/ = 1. , / 1 . =max1 || [1, ). - : 1 1 . . 21 , 23/2,
22, 23 2.
.
.
-
,
.. ( ). , , , .
conv() = { }
,
.
, conv() .
.. ( ). 1, , 1, ,
=1 = 1,
= 11 + + 1, , .
... . conv() .
:
= {
=1
, 0,
=1
= 1, , } .(.)
(i) . , = 11 + + = 11 + + , , , , 0
=1
= 1 =
=1
.
[0, 1] (1 ) + . ,
(1 ) + = (1 )(11 + + ) + (11 + + )= (1 )11 + + (1 ) + 11 + + .
-
, , (1 ) 0, 0
(1 )1 + + (1 ) + 1 + + = (1 )(1 + + ) + (1 + + )= (1 )1 + 1 = 1.
, (1 ) + . , .
(ii) . .
() = 2: T 1, 2 . , (1 )1+2 , .
() . 1, 2, , , +1 , 1, , +1 0
+1=1 = 1. 11 + +
+1+1 .i. +1 = 1
+1
=1
= +1+1 = +1,
.ii. +1 < 1
+1
=1
= (1 +1) (
=1
1 +1
) + +1+1,
, =1
1+1
(1 +1) (
=1
1 +1
) + +1+1,
.
-
,
, (.), . = 1
2 + 1
2
. . conv(). conv() , . , (ii), , conv() . , conv() . , conv() = .
.. ( Simplex). simplex (affine-). , . simplex 3, . 3 .
.: 3. .
.. (). (- ) . ..
.. (). . . conv() + 1 .
( Radon): conv() 1, , , 1, , > 0
=1 = 1 = 11 ++. 1, ,
. . 21, 31, , 1
-
. , 2, , 2(2 1) + + ( 1) = 0. ,
(2 3 )1 + 22 + + = 0.
1 = 2 3 1, 2, ,
(.) 1 + + = 0
(.) 11 + + = 0.
(.)
=
=1
(
=1
) =
=1
(
) ,
0. ( = ) , 1 . , . (.)
=1( (/)) = 1.
(/) 0 = 1, 2, , .
(.) , . / > 0 . /.
0,
= 1, 2, , . :
(i) = 0, 0 .
(ii) > 0, / /, /.
-
,
(iii) < 0, / 0 / (/) (/) 0,
. ... ,
.
:
= {(1, , +1, 1, , +1) 0,+1
=1
= 1, } ,
(+1)2 -
.
(1, , +1, 1, , +1) =+1
=1
conv()
. , () . , .
... :
(i) .
(ii) int .
(iii) 0 int() (0, 1) int().
(iv) int = int.
:
(i) , [0, 1]. (1 ) + . , , . , ., (1 ) + . ,
-
, (1 ) + (1 ) + . 2 0 2 0
((1 ) + ) ((1 ) + )2 0.
,
((1 ) + ) ((1 ) + )2= (1 )( ) + ( )2 (1 )( )2 + ( )2= (1 ) 2 + 2 (1 )0 + 0 = 0
(ii) int = , . int , , int(). (1 ) + int(). int(), 1 > 0 (, 1) . int(), 2 > 0 (, 2) . = min{1, 2} > 0 (, ) (, ) . ((1 ) + , ) (1 ) + int(). ( ..)
((1 ) + , ) = (1 )(, ) + (, ) ,
.
(iii) > 0 2(0, ) int() . (1/) conv(2(0, ), (1/)) , . ( .)
^
d
.: conv{, 2(0, )}. (1 ).
2(, (1 )) conv(2(0, ), (1/)) ,
-
,
int().
(iv) int() > 0 (, ) , (0, ) , 0 int(). (iii) (0, 1) () int(). , + int() + int(). , 1 int() : (0, 1)
+ + int.
+ int 1 .
, int int .
... .
conv() = conv().
: conv() conv(). .. (i), conv() conv().
, conv() conv(). . .. conv() , . conv() conv().
. ,
= {(0, 0)} {(, ) =1
, > 0} 2.
.. > 0
12
(, ) +12
(, ) = (0, ),
-
, [0, 1] ,
(1 )(, ) + (, ) = ((1 ) + , ).
.. (0, 1) (1 ) + = . 0 (0, 1) (1 0) + 0 = . (: .)
.. 1, , .
(conv({1, , })) = conv({1, , }).
, (conv()) = conv(()).
..
= { = (1, , )
=1
22
1}
1, , . () = 2
det 0. , 0 2 det 0.
.. kernel(),
kernel() = { [, ] }
. , =kernel().
.. . int() [, ) int().
.. simplex
conv({0, 1, 2 , }),
1, , , int() .(: > 0(0, ) 0 = (2)1
=1 .
(0, ) -,
=1
-
1. ,
= 11 + 22 + + + (1
=1
) 0
.) .. simplex + 1 . .. ,
conv( ) = 01
((1 ) + ).
.. conv() . .. simplex int() . .. ,
conv( + ) = conv() + conv().
..
conv(int()) = int(conv()).
.
, .
... , . :
= inf{ > 0 }.
, { > 0 } , 0 ( ..). > 0 (0, ) .
-
= 22/,
(0, ) = (0, ) .
:
(i) , 0 . = 0 0 > 0, 0 = 0. = 0 > 0 0 (/) . , . , > 0 /2 , , 2 0. = 0.
(ii) = 0 = || . 0,
= inf{ > 0 } = inf { > 0 }
( = /||)
= inf {|| > 0 ||
}
(||/ = 1 = )
= inf{|| > 0 }= || inf{ > 0 } = || .
(iii) > 0 infimum , , + > ,. , , > 0 , , < + .
+ , + , = (, + ,)
+ , +, < ++ + = + +2
> 0, + + .
..
-
. .
... . { 1} .
: = { 1}. = .
= inf{ > 0 } 1,
1 . . , 1.
(1 1/) 1 1/ < 1. (0, 1) (1 1/) . int() ( .. .. (iii)). , (1 1/) :(1 1/) 2 = 2/ 0 .
.. = = . ..
{ = (1, , )
=1
22
1} ,
=
=1
22
1/2
.
.
.. ( ). (dim = 1).
-
0 bd() 0 +.
0, 1. , 0, .
... . 1 ( bd()) .
: ( ..), , , , 0 . 0 . , , > 0 0 + 2 .
0 +2
0 + 2 ,
0 +2
, = 0, +2
> 0, ,
0. 0, 0, 0,
0., , 1,
1 1, =0, . , 1, < 0, 0 1, , 1, > 0, 1 0, .
1, = { , = 1, }
= { , = 0, } = 0,.
1 :
-
... . - 1
() = sup{, },
( .). .
.: 1.
... . > 0
+ = +
= ,
, .
:
+() = sup{ + , , , }= sup{, + , , , }= sup{, , } + sup{, , }= () + ().
-
, > 0
() = sup{, }= sup{, }= ().
... , ,
() (),
1.
: ()
() sup
, sup
, = ().
() 0 . () = 0 2
|()()| = 0202 (0)(0)2 = 2.
, 0 . 0 0 02 0, = (0 0)/0 02 1.: , 0, .[ : 0, > 0. 0 ( ).
(1 )0 + = 0 + ( 0).
, (0, 1)
0 (0 + ( 0))2
2= 0 0 ( 0), 0 0 ( 0)= 0 022 + 2 022 20 0, 0= 0 022 + ( 022 20 0, 0).
0 0, 0 = 02, 0 > 0,
-
0 (, 0 < < 20 0, 0/ 022)
0 (0 + ( 0))2 < 0 02
0. ]
, , , 0, . ,
(.) () = sup
, = 0, ,
0 . 0, 0} . (, ) = conv{1, 2, , } ( ) : 1, 2, , 1, 2, , 2 (, )
(, ) = conv{(1), (2), , ()},
(, ) 1, ()
:
0 () =
=1
( )2 = 0 + 1 + 22 + + 22,
1, , . = { , = 0} =(1, 2, , 2, 0, 0, , 0) . () .
... () -neighborly . -neighborly, 3 (4 ) -neighborly, simplices -neighborly. 3 2-neighborly 5 , 2 4. .
.. , {0}
()() = ().
..
{ = (1, , )
=1
22
1} ,
-
= (1, , )
() =
=1
2 2
1/2
.
.
... 0 ,
= { , 1, }.
.
( ) .
... . , , 1 , (, , , )
=
.
, , 1 ( .). ,
(.) = 1.
( [], .). F. J. Servois J. D. Gergonne.
-
z
x
y
x
y
z
.: 1.
( ) . (- ..) : ( ..). ( ) () = , .
.
(i) . , , [0, 1]
, (1 ) + = (1 ), + , (1 )1 + 1 = 1,
(1 ) + .
(ii) .
() int . , - . , > 0 2(0, ). -
-
,
2 (0,1)
.(.)
2(0, 1/) 2 1/. 2(0, ) 2 .
, 2
2
1
= 1.
(.). - 0 int. int .
() . , 0 int > 0 2(0, ) . 2(0, 1/). {0} /2 2(0, ) . , /2 1 2 1/.
() . , . , 2 0. , 1 . , 1 , , = lim, 1.
... .
: . , , .
, = , 1. ...
0 1, , .
= {(1, , )
=1
22
1} .
-
= {(1, , )
=1
2 2 1} ,
11 , , 1 . Cauchy-Schwartz. 11 , , 1 , = . = (1, , ) = (1, , )
, =
=1
=
=1
(
=1
22
)1/2
(
=1
2 2 )1/2
1.
., = (1, , ) , =
=1 1
= (1, , ) . = 0 . 0 = ()=1
=2
=1
2 2
,
= 1, , . , 1,
=1
2 2 1. .
... 1 ., (1 ) , 1 1 . 0 |0 | = max=1, , ||. |0 | = 0 = 0
.
|0 | 0 = 0 0 0 = 0/|0 |.
1
(1 ) , 1. , = |0 | max=1, , || 1, . (1 ) . , max=1, , || 1, = 1, , . 1
|, | =
=1
=1
|| || max=1, , ||
=1
|| 1.
, 1 (1 ).
-
.: 1
... . , 1 .
. .. 1.
... ,
() = sup{, }.
0 () = 2(/2).
, = { () 1}. , , = { 1} ( ..). , .
... 0 .
() = .
-
:
() = sup{, } = sup {
, } .
(.)
supremum 1. / - 1, . / , 1 , supremum 1. 1, > 1
sup {
, } < 1.
sup {
, } < 1,
.
= > 1
( ..).
.. ( ..) (2)
= 2 > 0 (2)
=12.
.. , 0 , () 0 (()) = ()1().
.. ,
=1
2 /2 1 0
=2
=1
2 2
,
-
= 1, , , . .. 0
(i) ( ) = conv( ).
(ii) ( + ) . .
.. 0 , > 0 ()(1) 2
2 +
1. .. 1 < < 1 < < 1+1 = 1 ( ) = . .. 0 .
(i) bd() .. . (: (.) { , = 1}.)
(ii) bd() bd().
.. 0 .
(i) bd() .. .
(ii) bd() bd().
.
.. (()) = 0 < < 1. 0 , () = .
-
... , ,
+,
+, ,.
, +, ,
, .
... , ,,, < , ,,
+,,
+,, ,,.
,, +,,
, < , ,
.
... , :
(i) ( ).
(ii) = { , } {0} ( ).
: (i)(ii): , 1 1, 2 2 [0, 1],
(1)(11)+(22) = ((1)1+2)((1)1+2) .
,, = { , }, < , . { , } { , }. ,
{ , }.
, < 0, {0}
,,0 = { , 0}.
(ii)(i):
,,0 = { , 0},
> 0, {0}. , , , , + , ,
-
. = sup{, }, + , , . ,
,,+ = { , + }. ... , .
, = , , .
: .. 0 , 0
(, ) = 0 02 > 0.
, = 0 0 . -
{ 0, , 0, }
. ,
, 0, 0, 0 0, 0 0 0.
, 1 1 0, 0 0 > 0. [0, 1] , ,
= 0 + (1 0) , [0, 1].
0 22 = (0 0) ( 0)22= (0 0) ( 0), (0 0) ( 0)= 0 022 + 022 20 0, 0= 0 022 + 21 022 20 0, 1 0= 0 022 + (1 022 20 0, 1 0)< 0 022
> 0 , .
-
,
{ , , 0}.
... . ...
.
... 0 int() () = .
: , 1 . ( ()) (). ().
, (). . {} . {0}
, 1 < , ,
. . , (), . () .
... 0 int(),0 int() > 0 :
(i) () =1
.
(ii) .
(iii) (()) =()1().
: (i) () , 1 . , 1 (1/).
(ii) . , 1 . , 1 . . .
, () (). .. .
-
(iii) (()) , 1 (). (), 1 .
(), = () = () = () = , ().
(()) () , ()1().
.. , , . 1 1 . .. , , , = 1, , . ( ) = () ( ) .(: .. = (()).) .. . .
(i) . (: . {} .. .)
(ii) = int() int(). 0 . . 0 0 . .. .., (i).
..
= { },
. .. conv()
conv( {})) = conv( {})
= .
-
.. , = bar(), = , = (1, , )
=
= , ,
()=1 .
(i)
, = ,
.
(ii) bar( bar()) = 0.
(iii) : bar() int().
(iv) ( bar()) = bar().
-
.
.. ( ). vol() - , = [1, 1] [, ] ( )
vol() =
=1
( ).
, vol() ||. ,
- , .
... = 1 2
int() int() =
,
vol() =
=1
vol().
... . . , [0, 1] [0, 1]
([0, 1/2] [0, 1]) ([1/2, 1] [0, 1])
([0, 1/2] [0, 1/2]) ([1/2, 1] [0, 1/2]) ([0, 1] [1/2, 1]).
-
. . ...
...
vol( + ) = vol() vol() = vol(),
, > 0.
, :
... , , Jordan -
sup{vol() } = inf{vol() }.
.
vol() , vol() ||.
... ( .. (v)).
... () = 1 () = 0 Jordan Riemann ,
vol() =
() =
((1, , )) 1 .
.. . . . Jordan , .
-
... .
: .: , , .[ : = (, ), .. > 0.
0 = [4
,
4]
.
0 0
diam0 = 2
4=
2
< .
,
(.) + 0 ,
+ 0 . H (.) 0 . + 0 , . , int( + 0). .
int( + 0).
, . , 1, ,
=1
int( + 0)
=1
( + 0) ,
= =1( + 0) . ]
, int() . int() = 0 int(). > 0. Jordan . (1 ). ..(iii) (1 ) int(). , ,
-
0 (1 ) 0 int ,
0 1
1 0.
0 + +
11
0.
:
vol( + 0) = vol(0) sup{vol(), } inf{vol(), },
, . + (1 )10 . , ..,
inf{vol(), } vol( +1
1 0) = vol(
11
0)
= (1
1 )
vol(0).
(.)
0 inf{vol(), }sup{vol(), } ((1
1 )
1) vol(0).
. 0 . 0 , , . , , > 0 . 0 vol(0) vol() = . (.)
0 inf{vol(), } sup{vol(), } ((1
1 )
1) .
.
-
.. .. .
(i) = 1, , = 1, , = 1, , .
(ii) =1 vol() =
=1 vol() =
=1( ) .
..
- Riemann ( ..).
... :
(i) 1 2 vol(1) vol(2).
(ii) 1, vol() = 0.
(iii) vol() = vol().
(iv) vol( + ) = vol().
(v) = vol(()) = vol().
: (v). (v) :
= [0, 1] [0, ] = {
=1
, 0 } .
,
() = { (
=1
) , 0 } = {
=1
(), 0 } .
-
,
22 = , = () = = = , = 1.
, 1.
, = () = () = = , = 0,
. , . , () 1, , . ,
vol(()) = 1 = vol().
, - supremum .
.
... vol(()) = | det | vol().
: . () = 1() - det . () = 1 (1()) = 1.
vol(()) =
()() =
(1())
=
()| det | = | det | vol().
.. Riemann , vol() = vol() vol( + ) = vol() > 0 .
-
.. () = vol()vol(). 0 (()) = ().
.. , , =
vol(()) =
()() =
() = vol().
.. [ simplex] - 1, 2, , , 0 = 0, = 1, 2, , span{1 , 1} = conv{0, 1, , }.
vol() =1!
=1
.
.. . , 1.
vol( + 12)vol( 12)
.
(: + 12, , .)
.. . (, )
(, ) = min{ 1, ,
=1
( + )}.
:
(i) vol() (, )vol(). vol(+) (, )vol( + ).
(ii)vol( + )
vol() 2(, ).
-
(iii)
(, ) vol( + 12)
vol( 12) 2
vol( + )vol()
.
(: .. .)
(iv) vol() = vol() (, )1/ 4(, )1/.
. 1
. 2.
1 < .
=
exp() = (
exp(||) )
.
=
(
)
=
(
0[,)()
(
))
=
0(
(
) (
[,)() )) .
[,)() = 1 () = 1.
-
1
.
=
0(
(
)vol( )) = vol( )
0+1
.
vol( )
0+1
= (
exp(||) )
,
vol( ) =(2
0
)
0+1
=(2 (1 +
1))
(1 +)
,
() =
01 .
. :
lim+
( + 1)2(/)
= 1.
1() 2()
1()1
1/ vol( )1/ 2()
11/
.
..
(i) vol(1 ) = 2/!.
-
(ii)
vol(2) =
{{{{
(/2)! ,
2 1(( 1)/2)!
! .
.
.. 1- 2 .
.. vol(2) = 1.
.. = conv{2, } .
2 vol() < .
.. vol(2) = 1 , vol1(2 )
lim
vol1(2 ) = 2 .
.. = { || 1}.
vol(2) 1 1
.
2 (!) . vol((/2)2) = 1/2
0.
.. . - > 0 vol(2) vol((1 )
2) ,
, .
Busemann-Petty, : , vol( ) < vol( ) vol() < vol()?
-
1
.. 1 , =[1/2, 1/2]. Keith Ball ( [Bal]) 1 vol1( ) 2. 2 Busemann-Petty . ..* , . ( ), 1
1
|, | () =2 ( 1+2 ) (1 +
2)
( +2 ),
Haar 1.
(1
|, | ())1/
+
(:
|, |||
2/2
(2).
vol1(1)(2)
(
0+1
2/2 ) (1
|, | ()) .
- 1 - = (1, 0, 0, , 0), ||
2/2/2 .)
-
.
... , - , (, ) :
(, ) = max {max
min
2, max min 2} .
z
{
NBYOz NJOO{ 9 Y 9
NBYO{ NJOOz 9 Y 9
.: Hausdorff - .
. , , (, ) = (, ). (, ) = 0
max {max
min
2, max min 2} = 0.
,
{max min 2 = 0max min 2 = 0
{ , min 2 = 0 , min 2 = 0.
,
{ , , {
.
-
, = .H ,
.
... , ,
(i) (, ) = inf{ 0 + 2, + 2}.
(ii) (, ) = max{|() ()| 1}.
: , , . (ii) :
(, ) = max{|() ()|, 1} max{|() ()| + |() ()|, 1} max{|() ()|, 1} + max{|() ()|, 1} (, ) + (, ).
..:
(i) 1 = max min 2. , min 2 1, , 2 1. 12,
+ 12 + 12.
+ 12 + (, )2.
, 2 = max min 2,
+ 22 + (, )2.
,
inf{ > 0 + 2, + 2} (, ).
, > 0, + 2 + 2. + 2. , 2
max
min
2 .
-
, max min 2 . (, ) . ,
(, ) inf{ > 0 + 2, + 2}.
,
(, ) = inf{ > 0 + 2, + 2}.
(ii) > 0 + 2 + 2 1
() +2() = () + 2() = () + ,
() () . , 1 () () , |() ()| , max1 |() ()| . infimum max1 |() ()| (, ).: ,
(, ) = max {max
min
2, max min 2}
= max
min
2.
, 0 0
(, ) = 0 02.
(, ) = 0 = = . (, ) 0
0 =0 0
0 0 1.
: 0 0 0.[ : , 1 1, 0 > 0, 0. 10, 0 > 0. = (1)0+1 .. > 0 0 2 < 0 02. ]
-
(0) =0, 0. , (0) = sup{, 0 }
(0) 0, 0.
,
(0) (0) 0, 0 0, 0 = 0 0, 0
=0 0220 02
= 0 02.
,
max1
|() ()| (0) (0)
0 02 = (, ). ... , 1, 2, ,
0 int(). Hausdorff > 0 0 0
(1 ) (1 + ).
: (, ) 0 > 0 0 = 0() 0 (, ) < . + 2 + 2. 0 int() > 0 2 .
+ 2 = +
2 +
= (1 +)
.
(1 )
+
= + 2 = +
2 +
,
( ..)
(1 )
.
-
> 0 <
(1 ) (1 + ),
0 ., (1 ) (1 + )
(1) (1+) ( ..), (1 ) (1 + ) | | . -, . .
... vol() , , .
: , 1, 2, , , (, ) 0. vol() vol(). Hausdorff , , 0 ( ). .. > 0 0 0
(1 ) (1 + ).
(1 )vol() vol() (1 + )vol(),
vol() vol() max{((1 + ) 1), (1 (1 ))} vol().
> 0 > 0 .
.. - .. -
-
Hausdorff ( > 0 0 int() (1 ) int().).
.
... .
: 1, 2, , . , = 2 > 0.: 1, 2, 1, 2,
(, ) 1
2min{,}.
[ :
11, 12, 21, 22,
(.) (, ) 1
2.
,
1 = 11, 2 = 22, , = ,
, min{, }. ,
(, ) 1
2min{,},
(.). . , 1, 2, (, ) 1/2. + 1 ,
(+1,, +1,) 1
2+1.
-
,
= 2
int(2 (,1
2+2 )).
, (, 1/2+2) .
, = 1, 2, 3, (, 1/2+2) , , . (, 1/2+2) {1, 2, }. , (, 1/2+2) 1, 2, +1,1, +1,2, .
(+1,, +1,) 1
2+1.
+1,. 1/2+2 +1,.
2 1
2+2.
, +1, 1/2+2, . ,
(,1
2+2 ) +1, .
, +1, 2 1
2+2.
2 = ( ) ( )2 2 + 2
1
2+2+
12+2
=1
2+1.
, +1,, +1,
2 1
2+1.
-
, +1,, +1,
2 1
2+1.
,
(+1,, +1,) 1
2+1. ]
=
=1
( +1
212) .
+ (1/21)2 . . + (1/21)2 . (1, 2) 1/2
1 +12
2 2.
(2, 3) 1/22
2 +1
222 3 ,
+1
22 +1,
(, +1) 1/2. ,
1 + 2 = 1 +12
2 +12
2 2 +12
2 = 2 +1
222 +
122
2
3 +1
222 = 3 +
123
2 +1
232 +
121
2
= +1
22 +
12
2 +1 +1
22.
, . , . ,
-
+ (1/21)2. ,
+1
212 + 2,
( 1+log2(1/)). +
2. (, ) . = int( + 2), ( ). + (1/21)2
(1 + 2) (2 +12
2)
,
=1
(( +1
212) )
=1
( +1
212) = .
, = int( + 2), + 2 + 2 2(, ) + 2 int( + 2).
=
=1
(( +1
212) ) = (
=1
( +1
212)) .
, , , , . , . , 0 (0 + (1/2
01)2) .
0 +1
2012 + 2.
0
0 +1
2012,
+1
212 0 +
1201
2 + 2.
-
, +
121
2 + 2.
, + 2 + 2., (, ) .
-
PREKOPA-LEINDLER
BRUNN-MINKOWSKI
. :
(i) .
(ii) - (Fubini).
(iii) supremum ( Lebesgue). , |(2, 5)| = 5 2 = 3 [2 + 1 , 5
1] (2, 5),
sup
[2 +1
, 5 1
] = sup
(5 1
(2 +1))
= sup
(3 2)
= 3.
. -
... , , 0 - 0 < < 1.
(.) ((1 ) + ) ()1()
,
() (
() )1
(
() )
.
-
- -
... () = ()1() Hlder = 1/(1 ) = 1/ (1
+ 1
= 1)
() (
(()1))
1
(
(()))
1
= (
() )1
(
() )
.
, . (6.1) , () (6.1) () = . Prekopa-Leindler Hlder.
.. .
... [0, )
() =
0{ () } .
: = { () } 0. [0,()]() = 1 0 (), () = 1 0. . , , :
() =
()
0
=
0[0,()]()
=
0
()
=
0{ () } .
-
-
..: . = 1 :
: , , ( ) 0 < < 1 (1 ) + || (1 )|| + || , | | ( Lebesgue).[ : |(1 )| = (1 )|| || = || | + | || + ||. , , . , , , , . , (, 0] [0, ). , 0. , 0 ( ) + ( ) , 0 ( ) + ( ). , {0}
| + | = | + ( + )| = ( ) + ( )
( ) ( ) |( )| + |( )| || + ||.
. ]
. = 1. (6.1) :
{ () } (1 ){ () } + { () },
() () , , ((1 ) + ) , (1)+ { () }. ,
{ () } (1 ){ () } + { () }.
.. :
(1 )
+
(
)1
(
)
,
-
-
- -
- ( ..). .
1 , , . = 1, 1 () = (, ), ., = (1 )1 + 2 , 1
((1 ) + ) (1())1
(2()).
1
() (1
1())1
(1
2())
.(.)
, , () = 1 () . (.) ((1 )1 + 2) (1)1(2).
= 1 -.
() (
() )1
(
() )
,
1
(, ) (
1
(, ) )1
(
1
(, ) )
,
.
.. .. [0, )
() =
0vol{ () } .
..
-
-
= ! vol().
> 0
= (1 +
)
vol().
.. [0, ) 0 < < log (, log ). (0, 1) () = { () > } Lebesgue, , .(: .., + 1 .)
. -
Brunn-Minkowski . . Prekopa-Leindler.
.. ( Brunn-Minkowski). , , . :
vol ((1 ) + ) vol()1vol(),
[0, 1].
: = = 1
() = {1, 0, () = {
1, 0, .
= vol() = vol(). = (1 ) + = . , , Prekopa-Leindler, () = 0 () = 0 Prekopa-Leindler. () 0 () 0 , (1 ) + (1 ) + ((1 ) + ) = 1. Prekopa-Leindler.
-
- -
,
( )1
( )
1 ( 1)1
( 1)
,
vol() vol()1vol(),
vol ((1 ) + ) vol()1vol(). .. ( Brunn-Minkowski).
, . :
vol( + )1/ vol()1/ + vol()1/.
, .
: -!
. . || vol()
=
||1/, =
||1/
=||1/
||1/ + ||1/
( || = 0 || = 0 , || 0 || 0). [0, 1] 1 = ||1//(||1/ + ||1/). || = 1 || = 1. Brunn-Minkowski:
(||1/
||1/ + ||1/
||1/+
||1/
||1/ + ||1/
||1/ ) 11 1 = 1.
(
||1/ + ||1/+
||1/ + ||1/ )
1,
-
| + |1/ ||1/ + ||1/. ... Brunn-Min-
kowski . , , .
- Brunn-Minkowski.( -)
.. Brunn-Minkowski . .. > 0 = 1, ,
vol()
=1
vol() .
-
det (
=1
)
=1
det() .
.
... - .
= lim0+
vol( + 2) vol()
,
.
... > 0
() = 1().
-
- -
: | | vol,
() = lim0+
| + 2| ||
= lim0+
+ (/)2 ||
= lim0+
+ (/)2 ||
= 1 lim0+
+ (/)2 ||/
= 1(),
, .
... vol(2) = vol(), ,
2.
: || vol Brunn-Minkowski
= lim0+
| + 2| ||
lim0+
(||1 + |2|
1 )
||
= lim0+
(||1 + |2|
1 )
||
= lim
0+
(||1 + |2|
1 )
1|2|
1
1= ||
1 |2|
1 = |2|
1 |2|
1 = |2|.
,
(2) = lim0+|2 + 2| |2|
= lim
0+
|(1 + )2| |2|
= lim0+
(1 + )|2| |2|
= |2| lim0+(1 + ) 1
= |2| lim0+(1 + )1
1= |2|.
, (2).
-
.. ( ). . =(2)
vol() vol(2).
: (|2|1 /||
1 ) |2|, :
|2|
1
||1
= (|2|
1
||1
)
|| = |2|.
,
(|2|
1
||1
) (2).
, ..
(|2|
1
||1
)1
(2),
|2|/|| 1. , || |2|.
.. .
(|||2|
)
1
(2
)
11
.
.
Brunn-Minkowski Brunn. .
-
- -
... 1. = { , = 0}
() = vol1( ( + )).
1/(1) .
: = . () = vol1( { = }). () = { 1 (, ) } () = vol1(()). , ( () 0 () 0) [0, 1]
((1 ) + ) (1 )() + ().
, = (1 ) + () (), (, ) (, ) (1 )(, ) + (, ) . ((1 ) + , (1 ) + ) , (1 ) + ((1 ) + ).
Brunn-Minkowski 1 ( | | 1-)
((1 ) + )1/(1)
(1 )() + ()1/(1)
(1 )()1/(1) + ()
1/(1),
. Brunn Brunn-
Minkowski. . - ...
.. Brunn Brunn-Minkowski .
(i) 1 = {0} +1, 1 = {1} +1
= conv{1, 1} = {(1)(, 0)+(, 1) [0, 1], , }.
() = (1 ) + [0, 1].
-
(ii) Brunn (0), (1) (1/2).
.
Borel 1/2.
... 1 vol( ) = > 1/2. > 1
vol( ()) (1
)(+1)/2
.
:
2
+ 1() +
1 + 1
.(.)
, ,
=2
+ 1 +
1 + 1
,
(), .
1 =
+ 12
+ 12
().
, . / . / , .
(.)
2
+ 1(() ) +
1 + 1
( ).
Brunn-Minkowski
1 = | | () 2/(+1)| |(1)/(+1),
-
- -
.
.. Borel =2/|
2|
1/ . .. = /||1/ = 1 /|1 |1/.
-
STEINER
0 . Steiner : . St() ( St() ), Steiner . .
.
Proj
.
Proj() =
, .
... 0 . Steiner
St() = { + Proj() || 12
| ( + )|} ,
, 2 = 1 = { }.
0 0, . ( ). St() .
Steiner.
... 2 - ( ) Steiner .
-
.. . .
d\ \
Z [
[
[
[ Y Z
\ Z
[
\
x
y
z
{
|
}
Z
YZ
[
Y[\d\
~
.: 1 2.
a priori , () . St() = . , , , .
: , , ( .). (0, 1). (1)+.
-
conv{{0} [/2, /2], {} [/2, /2]},
.. : {} [/2, /2] 1 2 = 2 1
=2
+ =
1
.
|| =
2
+ 1
.
(1 ) + , .
... () St( + ) = St() + Proj().
: = 1 . (, ) , 1 St(+) (, ) Proj
(+) || (1/2)|(+)(+)|.
Proj( + ) = Proj
() + Proj
().
( + ) ( + ) = (( ) + ),
. ( ) ()+ = Proj
()+. (, ) St(+)
Proj() Proj
()
|| 12
(( Proj()) + ).
( Proj(), ) St()
(, ) St() + Proj(). ... , St() .
-
: . > 0 2(0, ). St() 2(0, ), St() . . ,
St() = {(, ) Proj() || 12
| ( + )|} ,
1. , (, ) St() (, ) ( ) . Proj() , Proj
. ( ..). Proj().
|| 12| ( + )|
|| 12| ( + )|. (, ) St() St()
. .
2(, ) > 0. 2(0, ) . Steiner , 2(0, ) St( ). .. St( ) = St() Proj
() 2(Proj(), ) St(),
. St() . ,
St()
= conv (( ( + )) ( ( + )))
( .). , St() .. St() , , [, ] St() St(), St() , .
... 0
St() = St().
: 0.
-
z
TU
TUz
.: Steiner. .
,
St() = { + Proj() || 12
| ( + )|}
= { + Proj() || 12
(
+ )}
= { +
Proj()
||
12
(
+ )} .
-
= / = /. :
St() = { + Proj() || 12
| ( + )|}
= { + Proj() || 12
| ( + )|}
= St(). ... ,
St( + ) St() + St().
: St() St(). = + = + , , , ,
|| 12
( ( + )) || 12
( ( + )) .
, + = ( + ) + ( + ). + , +
(.) | + | || + || 12 (
( ( + )) + ( ( + ))) .
( + ) + ( + ) ( + ) ( + + )
Brunn-Minkowski
( + ) + ( + ) ( ( + )) + ( ( + )) .
(.)
| + | || + || 12
( + ) ( + + ) .
, + St( + ). ... , ,
St() St().
-
: (+) (+)
(.) | ( + )| | ( + )|.
St().
= (Proj) + || | ( + )|.
(.)
= (Proj) + || | ( + )|.
, St(). , St() St(). ... St
.
: , 1, 2, ,
St() St().
0 int(). .., > 0, 0 0
(1 ) (1 ).
,(1 )St() St() St()(1 + )
. ...
vol(St()) = vol().
: - 1 .
-
vol() =
((1, 2, , )) 1 1
=
( + (1, , 1)) 12 1
=
St() ( + (1, , 1)) 12
=
St()((1, 2, , )) 1 1
= vol(St()),
| ( + )| =|St() ( + )|.
...
(St()) ().
:
St( + 2) St() + St(2) = St() + 2
vol(St() + 2) vol(St())
vol(St( + 2)) vol(St())
=vol( + 2) vol()
.
(St()) (). ... diam(St())
diam().
: , St() , St() - .., 2. , diam(St()) diam().
-
... () = max{ >0 2 } () = min{ > 0 2},
(St()) () (St()) ().
: 2 2 St()
{ > 0 2 } { > 0 2 St()},
() (St()). , (St()) (). ... -
. Steiner ,
vol((St())) vol () .
: - = (0, 0, , 0, 1), 1 ( - ). (, ) 1
St() = {(,12
( )) (, ), (, ) } .
:
(.) = {(, ) , + 1 (, ) } ,
, 1 , . ,(.)
(St()) = {(, ) , +12
( ) 1, (, ), (, ) } ,
, 1 , , . () = { 1 (, ) }. , () {}
-
1.
12(
() + ())
=12 {
1 (, ) } +12 {
1 (, ) }
= {12
+12
(, ) , (, ) }
= {12
+12
, + 1, (, ) ,
, + () 1, (, ) }
( , + 1 , + () 1)
{12
+12
,12
( + ) +12
( ) 1, (, ), (, ) } .
= (1/2) + (1/2) :
12(
() + ()) { , +12
( ) 1, (, ), (, ) }
(St())(),
(.). Brunn-Minkowski 1 :
vol1((St())())
11 vol1 (
12
())1
1
+ vol1 (12
())1
1
= 2vol1 (12
())1
1
,
. ,
vol1((St())()) 21vol1 (
12
()) = 21 (12)
1
vol1(())
,vol1((St())()) vol1(()),
-
, ,
vol1((St())
())
vol1(())
,
vol((St())) vol().
.
Gross . Blaschke-Santal .
... vol() = vol(2). - - Steiner 0.
2,
Hausdorff.
: = inf{() }. 0, . infimum
(.) () .
, > 0 () , , () ( ) . , - , . , 0.: (0) = .
-
[ : 0, (, 0) 0, > 0, 0 0, (, 0) < . 0 + 2 0 + 2. 0 (0)2,
0 + 2 ((0) + )2 (0 + 2) (0) + .
,
( + 2) () + .
() (0 + 2) (0) + ,
(0) ( + 2) () + .
, () (0) (0) () . , |() (0)| , 0. ,
(.) () (0).
(.), (.) (0) = . ]
= (0) 0 2. - 0 = 2. , , 0 2, 1 0: , 1 0, 0 2 = conv(1) 0 . 0 1 0. 0 0 0. 1 0, 0 , 1 0 0 = . 0 0, : , 1 0 0 [, 0] ( .).
1 1
-
.: 0 0 1.
z
.: 0 .
1, . 1, , 1
1
=1
,
0 . -
1, 2, , .
0 = St(St1( St2(St1(0)) )).
-
0 int(2), - 2, 0 ( .).
z
jY
.: 0 .
, 0 int(2) (0) < . Hausdorff, 0
= St(St1( St2(St1()) )) St(St1( St2(St1(0)) )) = 0.
:() (0) < ., 0 (0) < 0 . , .
, Hausdorff,
vol(2) = vol() = vol(0) = vol(2).
, = 1. , 0 = 2. , 2.
-
( )
.. 1, 2,
= St(St1( St1()))
2 > 0. (: Gross 1 2 1. Gross, 1.) .. . 1, 2,
= St( (St2(St1())) )
= St( (St2(St1())) )
vol()1/2 vol()1/2 . (: -
.. 2
2 -
). .. ( Brunn-Minkowski) = 2 =
2
vol( + )1/ vol()1/ + vol()1/.
.. Brunn-Minkowski ( Prekopa-Leindler).
. ( )
Steiner.
|| = |2|. Gross, , 1, 2, , , , Steiner 0, 2, . (),
-
Steiner . . , .. St(2) = 2,
| + 2| = |St( + 2)| |St() + St(2)| = |St() + 2|.
Steiner , | + 2| | + 2|. , || = ||. ,
| + 2| ||
| + 2| ||
.
2 Hausdorff
| + 2| ||
|2 + 2| |2|
.
, 0+ ., (2). . -
Gross .
... 0 int()
vol()vol() vol(2)2.
.
: - .
.. , vol((St())) vol(). 1,2, 2 ( = (vol()/vol(2))1/ )
(.) vol() vol().
-
-
,
vol((St2(St1()))) vol((St1())
) vol(),
. 2 .. 0 < < 1
0 0
(1 )2 (1 + )2.
..
11 +
1
2 1
1 1
2.
0 < < 1 (1 )1 1 + 1 (1 + )1( = 21(1 + )1) 0
(1 )1
2 (1 + )1
2.
12. (.)
vol(1
2) vol().
vol() vol(2) ( ) .
vol()vol() -,
vol(())vol((())) = vol()vol().
, .. .., -
vol(()) = | det |vol() vol((())) = | det |1vol(),
-
. vol()vol() . . (4! /) simplex. Mahler. -
-
Jean
Bourgain
Vitali
Milman
??.
.. Blaschke-Santal .
vol()vol() = vol(2)2.
.. , , 0 , vol() = vol(). vol() vol().
-
II
-
. - [] [].
-
.
.
GL() , . GL() det 0.
() , = Id. () +1 1. () 1 = , :
(i) , = , = , ,
(ii) 2 = , = , = 2.
SL() , 1.
GL(), () SL() .
.. ( ). tr() . = ()
tr() =
=1
=
=1
, ,
()=1 ().
- = (), = () :
-
(i) tr() + tr() = tr( + ).
(ii) tr() = tr().
(iii) tr() = tr().
(iv) tr() = tr() = ,
( ..).
(iv) , tr(1) = tr(1) = tr. , , .
... - {0} , > 0.
... - {0} , 0.
. , ( []), 1, , 1, , . = 0 = 1, , . .
... = { , 1}.
: , 1, , 1, , ...
=
21 0 00 22 0 0 0 2
.
: = diag(21 , , 2 ).
-
, =
=1
, 2
2.
= { , 1}.,
= { , 1}. . 1 , 1, , , , . , 1. , 1 , =
=1, ,
(
=1
, ),
=1
, 1.
=1
, 2
(1/)2 1,
...
.. = () = () tr() = , . (: .. ()=1
= =1, .)
.. . 2
det(Id +) = 1 + tr() + (2),(.)
(2) lim0 (2)/ = 0. (.) tr - Id,
-
tr() = lim0
det(Id +) det Id
.(.)
(.) tr(1) = tr.
.. [Singular Value Decomposition] - , = . -
-?
(i) 1, , 1, , = = 1, , .
(ii) = 1 = 1, , , ta 1, ,
.
(iii) o
= diag(11
, , 1
).
.. .. = . . (: = ()() .)
.
. , 1 .
= sup{2 2}.(.)
, 2 . , (.) :
-
(i) 0 = 0 = 0 2, {0}
= 2 (
2) = 0.
= 0.
(ii)
= sup2
2 = || sup2
2 = || .
(iii) , ,
+ = sup2
( + )2 sup2
(2 + 2)
sup2
2 + sup2
2 = + .
... 1. -, ( ) = Id. ,
22 = , = , = , = 2.
2 = 2 (.) = 1.
... > 0 2 2 . (.).
2 2. , = 0, 0
2 = (/2)2 2.
... , . , (.) .., 2
()2 = ()2 2 2,
.
-
... 2
(),=1 = (11, 12, , 1, 21, , 2, , 1, , ),
, 1-1, , .
: , - . 1.
2
( ).
2,
... ...
(2 , 2) ( ..).
( ) (2 , 2)
.. = diag(1, 2 , ) ,
= max{|| = 1, 2, , }.
.. det() = 1 1 . .. .. .. , - det() = 1 .
-
. 0 + . . . . . . . GL() : GL() .
... 0 + () 0 GL().
( ). 2.
... 2 .
: 1, , 1, ,
= {
=1
, 2
2 1} ,
( ..). = diag(1, , ) - ( 1, , )
-
(2) = . .
, GL() (2) . (2) 1 2 1, 1 1. (1)1, 1. , .., (1)1 . ,
(1)1, = 1, 1 = 12 0.
{0} 1 0. 1 = 0 1 =/2 2, , 1. 1 1, , , 11 = 0 11, = 0 = 1, 2 , . det 1 = 0, , .
... : () = ( = 1) ( []). 2 ( ).
... ,
(i) GL() = (2).
(ii) .
(iii) ().
.
-
. ., 1, ,
= { = ()=1
=1
22
1} ,
> 0 = 1, , ,
, =
=1
2
(.)
{ , 1} .
.. 2 = [1, 1]2. = [1, 1]. .. (.) . .. , , = 2. .. .
-
. :
.. ( ). 1, 0 ,
2 , 1. .
GL() () .
. .
.
..
. . , , . . = , .
( ), ( ).
-
, .
.
. , , . . .
. . ( .). . , ( ) . ( o . !)
, . , = 22.( . , .)
, . 1. , 1. 1 , ()
-
.: Samuel Dixon / ( George Barker, ).
22 ,
, . ,
(22 , 2) .(.)
. , , . ,
2 1.
, .
, - .., 0
-
1, - .
..
. ; - ; .
Id .
=1
, (.)
(22 , 2) = , ,
(
22 ) Id (
)
=
(22 21 ) 12 1
21 (2 22) 2
1 2 (
22 2)
.
(Inertiamatrix). . ( ) 1 , . , ,
-
.
. - () 1
2, ,
. , = , 1, , . ( ) . 1, , ( , ) () . 1 ( ) .
- . , ( ) , , ,
2 = 0 .
, , !
.
.., 1 0.
= = Cov() ( )
() =tr() 1
Id .
,
tr() =
=1
(
22
2 ) = ( 1)
22 .
= (Covariance
-
matrix) . . , Id ( ) . :
.. ( , ( )). - 1, 0
(i) 2 =
2 , = 1, ,
(ii) = 0 , = 1, , .
= ( 2 )1/2 ( )
.
... ...
.. (i) (ii) .. ,
2 1.
..
- ( ). . 0 . - , 0. ,
-
0. 1 0 = 0
PI =
2 .
. +1 0.
:
PI =
22 = tr() =tr() 1
.(.)
..
. .
, ( ). .
, , , , .
22 =
=1
( )2
. . 2 , = 0, . , = 0 = , . . = ( ).
-
. , , ,
( ) ( ) = ( ) = = ( ) ( )= ( ) ( ).
, , = = . - . . ( ) . ( ) ( ).
=122,
. 1 = 1
2 2 .
= 1 = .
= (22 , ) = ,
=
22 21 12 121 22 22 2
1 2 22 2
.
= . = 1
2, .
-.
-
. = 1 = 1 . . .
, . , = 1
2, ,
.
, , - . : .
, .
, - . -, .
... , , , , , . . - . ,
-
..
, , 1/2. =
12, ,
, = 1. - Poinsot () . , -
=1
2 = 1,
. O Poinsot =
12,
= 2grad.
= , . Poinsot . , , - . =
12,
, . (invariable plane). - . ( polhode +). - ( herpolhode ++).
. ,
-
r
.: Poinsot, , . . , ( ) , , ( ) .
. -
=1
1 2 = 1.
Binet .
=1
2 = 22 , -
2 Binet. , Binet Poinsot.
... Poinsot , , Legendre, ,
-
. ... Binet Legendre. Legendre., Legendre. Binet , Legendre ( ..).
... polhode herpolhode . Principia Isaac Newton. Newton Leonhard Euler , . Jean d Alembert Louis Lagrange , .
Louis Poinsot - Poinsot , Euler. [GPS] [KO].
.. , Legendre ()
= { sup
(, ()) < }
()() = sup
(, ()).
( -)
-
,
(12
2) =12
2 .
Poinsot Legendre Binet,
(12
2Binet) =12
2Poinsot.
..* .. .
(12
2) =12
2 ,
0 ( ), :
(i) > 0 2 > , 12
2 .
, 0.
(ii)
(12
2) () 12
()
(12
2) () = , 0 12
02 max0 (,
00
12
2) .
(iii)
(12
2) () 12
(),
0 bd() () = , 0 0 = 0
(12
2) () max0 (, 0
12
2) .
: ( )() =
-
( 2 {0} ( [GH] , .)).
.
.
... GL() () .
:
() = , = (
, 1 )
=1
.
= ( ). , 0
(), = , 2 > 0.
2 = . = 1(). 1 (1). 1 = ,
, 2 = | det |1
1, 2
= | det |1 , (1)2
= | det |1 , 12
= | det |1, 1 , 1
= | det |1(1), 1= | det |1,
-
. : = vol(1())1/1().
(- ). .. .
... ,
tr()
(det )1/
.
: -. 1, , ( )
tr()
=1 + +
(1 )1/ = (det )1/.
.
... PI, . , GL() det() = 1,
22
22 .(.)
: (.) (.). -
, =
(
) = tr().(.)
(.) = Id ( = 2 Id):
22 = , = tr(2 Id) = 2.(.)
-
(.) (), - ( = 1)
()
22 = ()
, = , = tr().
(.)
= 2 Id, - - tr()/ det()1/ = 1( ..), (.)
()
22 = 2tr() 2 det()1/ =
22 ,(.)
. ...
. , .
... (.) .. . (.)
= (1
22 )1/2
.
(.) ..
= inf{{
(1
()
22 )1/2
det = 1}}
= inf{{
1
vol(())1+ 2
1
()
22
1/2
GL()}}
... - .
-
: () . , .. () . (.). - tr()/ det(), ( ..). 1, . = Id, .
(.):
...
22
22 ,(.)
1, .
: , .., , () , (.), , . .. Binet Legendre
[MP] V. Milman A. Pajor Binet Legendre- . () - . Binet
(.) 2Binet = ,
2.
Binet . Binet ,
-
()
() = (
22) Id ,
. , 1, 2, , , () = diag()=1 = diag()=1 . + =
22. (.)
2Binet = , 1
2 =
2 = =
22 .
Binet Poinsot , - . , :
2Binet =1
2Poinsot = .
Legendre Legendre,
(.) Legendre
, 2 =
,
2,
.
Binet = + 2
vol(Legendre)Legendre,
. -
2Legendre = + 2
vol(Legendre)1
.
, - [MP], . , ()
-
, 2. , .
... [MP]
Binet = + 2
vol(Legendre)Legendre.
: GL() = (2) = ()2 .
1vol()
,
2 =1
vol(2)
2
, ()2.
:
1vol(2)
2
, ()2 =
1, ()
2(
1
0+1) ()
=
+ 2
1, ()
2()
=1
+ 2()22 =
1 + 2
2 .
= + 2vol() (
,
2)1/2
.
Legendre (.) (.) .
.. GL() = (2)
1vol()
,
2 =1
vol(2)
2, ()
2,
-
. ... .. () /Poinsot . ()
PI(()) = vol(12 )vol(2)
vol(())
=1
1
,
()=1
=(3/2) ( +12 )
( +44 )
263/2
13/2
.
, ()
(()) = vol(12 )vol(2)
vol(())
1
. /Poinsot
Legendre.(: (())
() ( + []) 1/2 (1 2 )1/2, . ...)
.
, , . . ,
2 , 2() .
.,
-
. , , .
... [0, ) , [0, 1]
((1 ) + ) ()1().(.)
... Lebesgue , [0, 1] (1 ) + ,
((1 ) + ) ()1().(.)
() = () Lebesgue . .
... [0, ) .
: Prekopa-Leindler ( ..). , , Lebesque [0, 1] (1 ) + Lebesque ,
((1)+)((1 ) + ) ()()1()().
..
(1)+ ( )1
( )
,
(1)+
(
)1
(
)
.
((1 ) + ) ()1(),
-
. ( -
, [Borel]):
.. (Borel). Lebesque [0, ) = .
:
( ..).
, (.) Brunn-Minkowski( ..).
... [0, ) .
: = 0. 0 (0) > 0. 0
() 0 ()
., 1 > 0 (1) 1(0),
0()
. > 1 1 = (1 )0 + = (1 0)/( 0). ,
(1) (0)1().
() (0) ((1)(0)
)1/
(0)1/ (0)(0)/(10).
-
0
() = 1
0
() +
1
()
1
0
()1 + (0)
1
(0)/(10)
1 1
0
() + (0)
1
(0)/(10)
1
() + (0)
1
(0)/(10) .
. 0 () -
. ... .. -
. .. ( - ) . ( ) .
1 .
1 =
=
() ,
0 =
=
()
. -
() = 1,
-
,
() = 0,
. , ( ),
sup
() = 1.
.. ...
...
sup
() = 1,
() = 1,
() = 0,
, 2()
1.
,
...
sup
() = 1,
() = 1,
() = 0,
() = 0 ,
2 ()
= 1, ,
-
, Cov() = ( () ) - Id. :
...
= (
2 () )1/2
.
= (det Cov())1/2.(.)
> 0 > 0 () = () . :
() =
1
1
() = 0(.)
=
2 () =2
2 () (.)
. , 1 . . (.) (.), = ,
det Cov() =1
2det Cov(),
(det Cov())
1/2 = (det Cov())1/2 = .
-
sup
=
= (sup
)1/
= (sup
)
1/
.
(sup
)
1/
(det Cov())1/2 = (det Cov())
1/2 = .
:
... ,
= (sup
)
1/
(det Cov())1/2.
- :
... 0 - 0, det 0 > 0 .
: 0 () = 0. , - () = 1 = (
)1.
() = , () =
, () ,
. , = 2. =
=
(sup )1/(det )1/.
-
() = ( )/ sup
sup
= 1. ,
() = (()) = () = 1,
0 =
() =1
sup
() =
1sup
1 (
() ) = 0.
, 2() -
1 ..:
, 2() =1
sup
, 2() =
1, 2()
= (sup
)2/(det )2/
, 12()
= (sup
)2/(det )2/
, 1(), 1
= (sup
)2/(det )2/1, 1
= (sup
)2/(det )2/
.
..
1
(0)/(10)
.. , - 1
.
-
JOHN
.
... .
: , , .
= sup{vol() + }
, . , . 0 < < . () ()=1, + vol() . , 0 . 0 . Blaschke ( ..). . Hausdorff vol() = . . ()=1 = 1, , ( + ). = 1, , . > 0 , . > 0 = 1, , .
= {
=1
, 2
2 1} ,
()=1 . (1, , , 1, , )
-
(1) [, ]. . - (1, , , 1, , ) (1, , , 1, , ). lim = = 1, , 1, , , = 1 = , = 0 , .
= {
=1
, 2
2 1} .
= . int() < 1. ,
0 0 < 1. =1,
2/2 < 1 0.
=1,
2/2 1, . int() . ., int()
=1,
2/2 < 1. , lim = lim = = 1, , , 0 0
=1 ,
2/2 < 1 0. . int() , , ..
... - .
: 1 +1 2 +2 , vol(1) = vol(2) = . 1, 2 () 1 = 1(2) 2 = 2(2). =
12(1 + 2) +
12(1 +
2)(2).
=12(
1 + 1(2)) +12(
2 + 2(2)).
vol() . Brunn-Minkowski
-
vol() (12vol(1 + 1(2))
1/ +12vol(1 + 1(2))
1/)
.
, ,
vol (1(2) + 2(2))1/ = vol(1(2))
1/ + vol(2(2))1/.
Brunn-Minkowski. .. 1(2) 2(2). . 1(2) = 2(2) 12 1(2) = 2.
2 + 2(2) 12 2 + 2 12 (). 1 + 1(2) 12 1 + 12 1(2) 12 (). 12 1(2) = 2 12 1 + 2 12 (). 1 = 12 1 2 = 12 2 1 + 2 2 + 2 12 (). 0 + vol() > vol(2), 20 + 2() 2, .
1 + 2 2 + 2 12 (). 1 = 2. . 12 () [1, 2]. 12 () [1, 2] . [1, 2] [, ] > 0 = 1.
0 ={{
, 12
(1 + 2)2 +
=2
, 2 1}}
.
vol(0) = (1+/2)vol(2) > vol(2) 0 12 (). 1 +2, 2 +2 12 (),
bd(0) conv ((1 + 2) (2 + 2)) .
-
0 bd(0). 0, 11 [, ] [0, 1] 0, 11 = (1 )() +
0 = 0, 11 +
=2
0,
= (1 ) ( +
=2
0, ) + ( +
=2
0, ) ,
conv((1 + 2) (2 + 2)),
=2
0, 2
2
=
=2
0, 2 022 = 1.
0, 11 [, ]. - 0, 1 > > 0. 0 + 2. ( 0, 1 < < 0 0 + 2.)
0 2 1. 0 bd
=2
0, 2 = 1 0, 12
(1 + /2)2,
0 22 = (0, 1 )2 + 1
0, 12
(1 + /2)2.
1
(0, 1 )2
0, 12
(1 + /2)2 0.
0, 1 - ( )2 2/(1 + /2)2. [, 1 + /2] 0, 1, .
-
.
... John 2.
John 2. ( , ). .
.. (John). John.
2 2.
: 2. 02 , 0 02 > . 0 0
conv(0, 2) .
= conv(0, 2) vol(2) 2 .
, 0 = 1 > .
= { 212
+
=2
22
1} ,
> 1 0 < < 1 vol() >vol(2) . . 0. 0 22 ,
=1
1 1
2
1
1 1
2
.(.)
-
eeY
x Y eYyY
.: = conv(0, 2) .
= (1, , ) bd(). > 0 0 < < 1 .
1 : 1 > 1/ () (1, (
=2
2)1/2)
(.),
(
=2
2)1/2
1
1 1
2
11
1 1
2
.
, bd() (=2
2)1/2 = 1 21 /2,
1 212
1
1 1
2
(1 1 )
.
(.)
2
2+ 2 (1
12 )
1.(.)
-
2 : 0 1 1/ 2. 2 1
12 (
1 2
2 )+ 2 1.(.)
=
> 1 0 < =
1 1/1 1/2
< 1(.)
(.) (.). . 1 > 1 2 , .
.. (.) (.) (.).
-
.
. -
Hlder : - (
) = 1, , , =1 1/ = 1
=1 1(
)
(
=1
())
=1
.
= 1/ = =
1/ ,
( 1()) :
(
=1
())
=1
(
() )
( ..). :
1(), , ,
.
. Brascamp-Lieb .
.. (Brascamp-Lieb, ). , > 0, = 1, 2, ,
=1 = 1(
) 0.
-
( ) = 1, 2, ,
(
=1
()) 1/2
=1
(
() )
,
= inf{{
det (=1
)
=1(det )
}}
,
( 1/2 = = 0).
... Hlder, 1., = = Id = 1, , .
= inf{{
det (=1 )
=1(det )
}}
1,
... = Id,
=1 = 1. , infimum, 1.
Brascamp-Lieb Hlder.
Prekopa-Leindler Hlder ( ..), , , Barthe, Brascamp-Lieb.
Prekopa-Leindler , , 0 - 0 < < 1,
((1 ) + ) ()1()
,
() (
() )1
(
() )
-
-
( .. ). ( ..) :
.. (Prekopa-Leindler, ). ,1, , [0, ) 1, , > 0
=1 = 1. 1, ,
(
=1
)
=1
(())(.)
()
=1
(
() )
.
0 , , (.) ( , , ). ( Brascamp-Lieb). Barthe Brascamp-Lieb. Brascamp-Lieb Barthe.
... , > 0, = 1, 2, ,
=1 = 1(
) 0. ( ) = 1, 2, , (Brascamp-Lieb)
(
=1
()) 1/2
=1
(
() )
.
(Barthe: Brascamp-Lieb) , 1() 0
-
(
=1
)
=1
() ,(.)
