TU Berlin Convex Geometry II Exercise sheet 11...TU Berlin Martin Henk Convex Geometry II WS 2015/16...

TU BerlinMartin Henk

Convex Geometry II

WS 2015/16

Exercise sheet 11

Discussion: Thursday, 11.02.2016.

Exercise 11.1 Let P be a polytope with (symmtric) facet data {(ui, φi), (−ui, φi) :1 ≤ i ≤ m}. Show that P is centrally symmetric (w.r.t. some point).

Exercise 11.2 Let K,L ∈ Kn. The set

K ∼ L = {x ∈ Rn : x + L ⊆ K}

is called Minkwoski difference of K and L. Show that

K ∼ L = {x ∈ Rn : 〈u,x〉 ≤ h(K,u)− h(L,u) for all u ∈ Sn−1}.

Is (in general) h(K,u)− h(L,u) the support function of K ∼ L?

Exercise 11.3 For K,B ∈ Kn, let r(K,B) = max{r > 0 : K ∼ r B 6= ∅}be the inradius of K w.r.t. B and we set

Kρ :=

{K + ρB, ρ ≥ 0,

K ∼ (−ρ)B, −r(K,B) ≤ ρ ≤ 0.

Show that (1− λ)Kρ + λKσ ⊆ K(1−λ)ρ+λσ.

Exercise 11.4 Let K,L ∈ Kn, 0 ∈ K ∩ L, p ≥ 1, and let

f(u) := [h(K,u)p + h(L,u)p]1/p .

Show that h(Kf ,u) = f(u) for all u ∈ Sn−1 where Kf is the Wulff-shapew.r.t. f .


Convex Geometry II

WS 2015/16

Exercise sheet 10


Exercise 10.1 Let K ∈ Kno , t > 0. Show that

N (K, tBn) ≤ N (K, 4tBn) N (Bn, (t/16)K?),

N (Bn, tK) ≤ N (Bn, 4tK) N (K?, (t/16)Bn).

Hence, e.g., if Bn ⊆ 4K then N (Bn,K) ≤ N (K?, 116Bn).

There is a theorem by Artstein-Milman-Szarek saying that there existsabsolut constants α, β > 0 such that for all K ∈ Kno

N (Bn, α−1K?)

1β ≤ N (K,Bn) ≤ N (Bn, αK

?)β.

Exercise 10.2 Let K ∈ Kno , r = (vol (K)/vol (Bn))1/n and let N (K, rBn) ≤ec n for an absolute constant c. Then K is in M-position with constant c.

Exercise 10.3 Let K ∈ Kn be centered, i.e., the centroid is at the originand let K − K be in M-position with constant C. Then K ∩ (−K) is inM-position with constant c(C) depending only on C.

Exercise 10.4 There exists an universal constant c1 > 0 such that for K ∈Kn with 0 ∈ K we have

vol (K)vol (K?) ≥ cn1vol (Bn)2.


Convex Geometry II

WS 2015/16

Exercise sheet 9


Exercise 9.1 Let v1, . . . ,vn ∈ Rn and | · | be a norm. Show that

2−n∑

ε∈{−1,1}n

∣∣∣∣∣n∑i=1

εivi

∣∣∣∣∣ ≥ max{|v1|, . . . , |vn|}.

Exercise 9.2 Let K,L ∈ Kn. For v ∈ int ((K − L)/2) show hat

ψ(v) = vol ((−v +K) ∩ (v + L))

is log-concave.

Exercise 9.3 Let K ∈ Kno and t ∈ Rn. Show that

γn(t +K) ≥ e−‖t‖2/2γn(K).

Exercise 9.4 (Concentration of volume in convex bodies) Let K ∈Kn, vol (K) = 1, and L ∈ Kno such that vol (K ∩ L) = r > 1/2. Then fort ≥ 1

vol (K ∩ (t L)c) ≤ r(

1− rr

)(t+1)/2

,

where Ac = Rn \A for a set A ⊂ Rn.First show that

2

t+ 1(t L)c +

t− 1

t+ 1L ⊆ Lc.

and then ask Brunn or Minkowski or both.


Convex Geometry II

WS 2015/16

Exercise sheet 8


Exercise 8.1 Let v ∈ Sn−1, t ∈ [0, 1] and let U ={u ∈ Sn−1 :

∣∣〈v, u〉∣∣ ≤ t}.

Then µ(U) ≥ 1− 4e−nt2/4 and, in particular, for t ≥ 4/

√n it holds

µ(U) ≥ 0.9.

Exercise 8.2 There exists always a δ-net on Sn−1 consisting of at most(4/δ)n points.

Exercise 8.3 For K ∈ Kno let M(K) =∫Sn−1 |u|K dµ(v).

i) Show thatM(K)M(K?) ≥ 1.

ii) Show that M(Bpn) ≥ (

√2/π)n1/p−1/2 for 1 ≤ p <∞.

Rem.: For ii) Holder and the next inequality might/could help.

Exercise 8.4 Let κn = vol (Bn) = πn/2/Γ(n/2 + 1). Show that√2π

n≥ κnκn−1

≥√

2π

n+ 1.

Show first that γn = κnκn−1

is a decresing function in n and consider γn γn−1.

———————————————————————-7.teUbung


Convex Geometry II

WS 2015/16

Exercise sheet 7


Exercise 7.1 Show that Kn equipped with log dBM(·, ·) is a compact metricspace, and that for K,L ∈ Kno

dBM(K,L) = dBM(K?, L?).

Exercise 7.2 Let Bpn = {x ∈ Rn :

∑ni=1 |xi|p ≤ 1} with B∞m = [−1, 1]n.

Show that for 1 ≤ p ≤ q ≤ 2 or 2 ≤ p ≤ q ≤ ∞

dBM(Bpn, B

qn) = n1/p−1/q.

The Minkowski-sum of finitely many line segments is called a zonotope Z,i.e.,

Z =

m∑i=1

conv {vi,wi}.

Exercise 7.3 Let Z be a zonotope. Then there exists a c ∈ Rn and ui ∈ Rn,1 ≤ i ≤ m, such that

Z = c +m∑i=1

conv {−ui,ui}.

For Z as above show that

vol (Z) = 2n∑

J⊆[m],|J |=n

|det(uj : j ∈ J)|.

Exercise 7.4

i) Show that the projection body (see Exercise sheet 4) of a polytope is azonotope.

ii) What is the projection body of the cube [−1/2, 1/2]n?

iii) What is the projection body of an ellipsoid?


Convex Geometry II

WS 2015/16

Exercise sheet 6


Exercise 6.1 Let K,L be 2-dimensional o-symmetric convex bodies. Showthat the inequality vol (K) ≤ vol (L) is a consequence of either of the nexttwo properties

i) vol 1(K ∩ lin {u}) ≤ vol 1(L ∩ lin {u}) for all u ∈ S1,

ii) vol 1(K|lin {u}⊥) ≤ vol 1(L|lin {u}⊥) for all u ∈ S1.

Is symmetry needed?

Exercise 6.2 Let K ∈ Knc and let c1, c2 ∈ R>0 such that for all u ∈ Sn−1

c1 ≤∫K〈x,u〉2 dx ≤ c2.

Show that√c1 ≤ LK ≤

√c2.

Exercise 6.3 Let K ⊂ Rn, L ⊂ Rm be two convex bodies in isotropic posi-tion, and let

M =

(LLLK

) mn+m

K ×(

LKLL

) nn+m

L ⊂ Rn+m.

Show that M is in isotropic position and LK×L = Ln

n+m

K Lm

n+m

L .

Exercise 6.4 Let a ∈ Zn, a 6= 0, gcd(a) = 1 and let S(a) = {〈a, z〉 : z ∈Nn≥0}. For x ∈ Rn let |x|0 = #{xi 6= 0 : 1 ≤ i ≤ n}.

i) Let ‖a‖ < 2n−1 and α ∈ S(a). Then there exists a z ∈ Nn≥0 with〈a, z〉 = α and |z|0 < n.

ii) Let α ∈ S(a). Then there exists a z ∈ Nn≥0 with 〈a, z〉 = α and|z|0 ≤ c log2 |a|∞, where c is an absolute constant and |a|∞ is themaximum norm.

Exercise 4.4 could be helpful.


Convex Geometry II

WS 2015/16

Exercise sheet 5


For K,L ∈ Kn let

N (K,L) = min{|S| : S ⊂ Rn with K ⊆ S + L}N (K,L) = min{|S| : S ⊂ K with K ⊆ S + L}

N (K,L) is called the covering number of K by L.

For K,L ∈ Kn, L = −L let

M (K,L) = max{|S| : S ⊂ K with |xi − xj |L > 1 for all xi 6= xj ∈ S}.

M (K,L) is called the separation number of K by L.

Exercise 5.1 Show that

i) for K,L,M ∈ Kn, K ⊆ L: N (K,M) ≤ N (L,M), N (M,L) ≤N (M,K), and N (M,L) ≤ N (M,K)

ii) for K,L ∈ Kn: N (K,L− L) ≤ N (K,L) ≤ N (K,L).

iii) for K ∈ Kn, λ > 0: N (K,λBn) = N (K,λBn).

iv) for K,L,M ∈ Kn: N (K,L) ≤ N (K,M) N (M,L).

Exercise 5.2 Let E1, E2 ∈ Kn ellipsoids centered at the origin. Show that

N (E1, E2) = N (E2?, E1?).

Exercise 5.3 Let K,L ∈ Kn. Show that

N (K, (K −K) ∩ L) = N (K,L).

Exercise 5.4 Show that

M (K, 2L) ≤ N (K,L) ≤ N (K,L) ≤ M (K,L)

Exercise 5.5 Let K,L ∈ Kn, dimK,dimL = n. Show that

vol (K)/vol (L) ≤ N (K,L).

If L = −L then

N (K,L) ≤ 2nvol (K + L/2)/vol (L).

Exercise 5.6 Let K ∈ Knc be in isotropic position. Then

c1 LK ≤ r(K) ≤ R(K) ≤ c2 nLK ,

where c1, c2 > 0 are absolute constants.


Convex Geometry II

WS 2015/16

Exercise sheet 4


Exercise 4.1 Show that among all n-simplices of inradius 1 the regularsimplex has minimal volume.The inradius is the maximal radius of an n-dimensional ball contained in abody.

Exercise 4.2 Show thatn∏i=1

xxii

attains its minimum on the set {x ∈ Rn≥0 :∑n

i=1 xi = α}, α > 0, if x1 =x2 = · · · = xn.

Exercise 4.3 Let K ∈ Kn. The set

Π(K) ={x ∈ Rn : |〈u,x〉| ≤ vol n−1(K|u⊥), for all u ∈ Sn−1

}is called the projection body of K. Show that

i) h(Π(K),u

)= vol n−1(K|u⊥).

ii) Π(AK) = |detA|A−ᵀΠ(K) for A ∈ GL(n,R), and Π(t +K) = Π(K)for t ∈ Rn.

Exercise 4.4 (Bombieri&Vaaler) Let a ∈ Zn, a 6= 0, gcd(a) = 1. Showthat there exists a z ∈ Zn \ {0} with 〈a, z〉 = 0 and max1≤j≤n |zj | ≤‖a‖1/(n−1).What could help is i) Minkowski’s theorem, saying that every o-symmetricconvex body with volume not less than 2n det Λ contains a non-trivial latticepoint of a lattice Λ, and ii) the set {z ∈ Zn : 〈a, z〉 = 0} is an (n − 1)-dimensional lattice of determinant ‖a‖.

Exercise 4.5 (McMullen) Let L be a k-dimensional linear subspace withorthogonal complement L⊥. Show that

vol k(Cn|L) = vol n−k(Cn|L⊥).

For a zonotope Z =∑m

i=1 conv {0,vi} the volume is given by

vol (Z) =∑

J⊆[m],|J |=n

|det(vj : j ∈ J)|.


Convex Geometry II

WS 2015/16

Exercise sheet 3

Discussion: Thursday, 12.11.2015!!

A function f : M ⊆ Rn → R>0 is called log-concave if ln f is a concavefunction. It is called centered if

∫M x f(x) dx = 0.

Exercise 3.1 Let K ∈ Kn be a centered convex bdoy, i.e.,∫K xdx = 0,

and let L ⊆ Rn be a k-dimensional linear subspace with orthogonal subspaceL⊥. For y ∈ relint (K|L) let f(y) = vol n−k(K ∩ (y + L⊥)). Show that f isa centered log-concave function.

Exercise 3.2 Let F,G : Rn → R>0 be integrable log-concave functions.Then, their convolution

(F ? G)(x) =

∫RnF (x− y)G(y) dy

is also a log-concave function.The Prekopa-Leindler inequality might help.

Exercise 3.3 Let K,L ∈ Kn with 0 ∈ intK, intL, and let λ ∈ (0, 1). Showthat

i)

vol (K?) =1

n!

∫Rn

e−h(K,x) dx.

ii)ln vol [λK + (1− λ)L]? ≤ λ ln volK? + (1− λ) ln volL?.

Here the Holder inequality might help

Exercise 3.4 Let Q ⊂ Rn−1 × {0} be a polytope, y ∈ Rn, yn 6= 0, andP = conv {Q,y} be the pyramid over Q with apex y. Let c(P ) be the centroidof P , and let q ∈ Q be the intersection of Q with the ray y + λ(c(P ) − y),λ ≥ 0. Then

|y − c(P )| = n|c(P )− q| = n

n+ 1|y − q|.

Exercise 3.5

i) Let P ∈ Kn be an n-dimensional polytope, and let Qi ⊂ P , i ∈ I, bepolytopes forming a subdivision of P , i.e., ∪i∈IQi = P and dim(Qi ∩Qj) ≤ n − 1, i 6= j. Find a relation between the centroids c(Qi) andc(P ).

ii) Let S = conv {v0, . . . ,vn} be a n-dimensional simplex. Show thatc(P ) = 1

n+1

∑ni=0 vi.

Exercise 3.6 Let K ∈ Kn mit c(K) = 0.

i) For u ∈ Rn let w(K,u) = h(K,u) + h(K,−u). Show that

1

n+ 1w(K,u) ≤ h(K,u) ≤ n

n+ 1w(K,u).

ii) Show that −K ⊂ nK.


Convex Geometry II

WS 2015/16

Exercise sheet 2


Exercise 2.1 Let H(a, 0) be a 0 containing hyperplane and let K ∈ Kn besymmetric with respect to H(a, 0). Let v ∈ Rn \ {0} ∈ H(a, 0). Show thatstH(v,0)(K) is still symmetric to H(a, 0).

Exercise 2.2 Similar to the first part of the proof of the Rogers-Shephardinequality show that for K,L ∈ Kn holds∫

Rnvol (K ∩ (x− L)) dx = vol (K)vol (L),

and conclude that that there exists a x ∈ 2K such that

vol (K ∩ (x−K)) ≥ 2−nvol (K).

Hence every convex body K of vol (K) = 2n contains a centrally symmetricsubset of volume at least 1.

Exercise 2.3 Let K ∈ Kn containing 0 in the interior and let h(K, ·) be itssupport function. The function ρ(K, ·) : Rn \ {0} → R≥0 given by

ρ(K,u) = max{ρ ≥ 0 : ρu ∈ K}

is called radial function of K. Show that

h(K,u)ρ(K?,u) = 1.

Exercise 2.4 Let K ∈ Kn, K = −K, dimK = n, and let L ⊂ Rn be ak-dimensional linear subspace with orthogonal complement L⊥. Show that(

n

k

)−1≤ vol (K)

vol k(K ∩ L) · vol n−k(K|L⊥)≤ 1.

For the lower bound it might be useful to observe that for x ∈ (K|L⊥)a suitable translation of (1 − ρ(K|L⊥,x)−1)(K ∩ L) + x is contained inK ∩ (x + L).

Exercise 2.5 Cauchy’s surface are formula states that for K ∈ Kn

F(K) =1

vol n−1(Bn−1)

∫Sn−1

vol n−1(K|u⊥) du,

K|u⊥ denotes he orthogonal projection of K onto the hyperplane with normalu. Use it in order to show

F

(1

2(K −K)

)≥ F (K).


Convex Geometry II

WS 2015/16

Exercise sheet 1


Exercise 1.1 Let K,Ki ∈ Kn, i ∈ N, with 0 ∈ intK. Show that Ki → Kif and only if for all ε > 0 there exists an iε ∈ N such that for all i ≥ iε

(1− ε)Ki ⊆ K ⊆ (1 + ε)Ki.

Exercise 1.2 Let K ∈ Kn. Show that the Steiner-symmetral stH(K) of Kwith respect to a hyperplane H is a compact set.

Exercise 1.3 Show that the minimal width of a convex boy may descreaseor increase under Steiner-symemmetrizations.

Exercise 1.4 Let K ∈ Kn. Show that (without using Exercise 1.5)

vol (K) ≤(

D(K)

2

)nvol (Bn).

Exercise 1.5 Let K ∈ Kn. The functional

w(K) =1

nvol (Bn)

∫Sn−1

[h(K,u) + h(K,−u)] du

is called the mean width of K. Here the integration “ du” is meant withrespect to the (n−1)-dimensional Hausdorff-measure. Let H be a hyperplane.Show that

w(stH(K)) ≤ w(K),

and conclude ”Urysohn’s inequality”

vol (K) ≤(

w(K)

2

)nvol (Bn).

Rem: The mean width is a continuous functional on Kn.

one more on the next page...

Exercise 1.6 Let T be an n-dimensional simplex, i.e., T = conv {v0, . . . ,vn},vi ∈ Rn, and dimT = n.

i) Let Hi,j, i 6= j, be the hyperplane through 12(vi + vj) and with nor-

mal vector vi − vj (orthogonal to the edge conv {vi,vj}). Show thatstHi,j (T ) is again an n-dimensional simplex.

ii) Show that among all n-dimensional simplices T the ratio R(T )/r(T )becomes minimal for a regular simplex. Here it might be helpful to usethe fact the surface area of the Steiner symmetral stH(K) is strictlydecreasing if K is not symmetric to the hypeprlane H.

TU Berlin Convex Geometry II Exercise sheet 11...TU Berlin Martin Henk Convex Geometry II WS 2015/16...

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