Some Geometrical Properties of Outer γ-Convex Sets

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Some Geometrical Properties of Outer γ-Convex SetsHoang Xuan Phu aa Institute of Mathematics , Hanoi, VietnamPublished online: 31 Aug 2006.

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MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016

NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION

Vol. 24, Nos. 3 & 4, pp. 303–309, 2003

Some Geometrical Properties of Outer g-Convex Sets

Hoang Xuan Phu*

Institute of Mathematics, Hanoi, Vietnam

ABSTRACT

A subset S of some normed linear space X is said to be outer �-convex w.r.t. some

given � > 0 provided that for all x0; x1 2 S there exist x�1; x�2

; . . . ; x�jwhich

belongs to S and the segment ½x0;x1� connecting x0 and x1 such that

kx�i� x�iþ1

k � � for i ¼ 0; 1; . . . ; j, where x�0¼ x0 and x�jþ1

¼ x1. This article

is devoted to some geometrical properties of such a set. For instance, if S is closed

and outer �-convex then any y 2 convSnS lies in some simplex whose vertices

belong to S and whose diameter does not exceed �. This implies that

convS S þ �BBð0; rÞ for r ¼ ð1=2Þ JsðXÞ �, where �BBð y; rÞ ¼ fx 2 X : kx � yk � rg.

As consequence, if S is outer �-convex and if x 2 X satisfies �BBðx; rÞ \ S ¼ ; for

some r > ð1=2Þ JsðXÞ � then x 62 clðconvSÞ, and therefore, some non-zero contin-

uous linear functional strictly separates x and S.

Key Words: Generalized convexity; Outer �-convex set; Simplex property;

Nonconvexity measure; Separation theorem; Self-Jung constant.

Mathematics Subjects Classification: 52A01.

*Correspondence: Hoang Xuan Phu, Institute of Mathematics, P.O. Box 631–Bo Ho, Hanoi,

Vietnam; E-mail: [email protected].

303

DOI: 10.1081/NFA-120022924 0163-0563 (Print); 1532-2467 (Online)

Copyright & 2003 by Marcel Dekker, Inc. www.dekker.com

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1. INTRODUCTION

Let X be a normed linear space. For x0; x1 2 X , denote

x� :¼ ð1� �Þx0 þ �x1; ½x0; x1� :¼ fx� : � 2 ½0; 1�g:

A subset S X is said to be outer �-convex w.r.t. some given � > 0 provided that

for all x0; x1 2 S there exist x�1; x�2

; . . . ; x�j2 S \ ½x0; x1� such that

kx�i� x�iþ1

k � � for i ¼ 0; 1; . . . ; j; where x�0¼ x0; x�jþ1

¼ x1:

The latter condition means

�0 ¼ 0; �jþ1 ¼ 1; 0 � �iþ1 � �i ��

kx0 � x1kfor i ¼ 0; 1; . . . ; j:

This notion was introduced in Phu and An (1999), in relation with so-called outer�-convex functions, whose class is very large but still maintains some crucialproperties w.r.t. global optimization.

Some basic properties of outer �-convex sets were already investigated in Phuand An (1999). This article is devoted to some further geometrical properties. Thefirst one is the following simplex property: If S X is closed and outer �-convexthen any y 2 convSnS lies in some simplex whose vertices belong to S and whosediameter is not greater than � (Theorem 2.1). This implies that the nonconvexitymeasure of such a set does not exceed ð1=2Þ JsðXÞ �, where JsðXÞ is the self-Jungconstant of X, i.e., convSS þ �BBð0; rÞ if r ¼ ð1=2Þ JsðXÞ �, where �BBð y; rÞ ¼fx 2 X : kx � yk � rg (Theorem 3.1). As a consequence, we have the followingseparation theorem: If S is outer �-convex (and not necessarily closed) and�BBðx; rÞ \ S ¼ ; for some r > ð1=2Þ JsðXÞ � then x 62 clðconvSÞ, and therefore, somenon-zero continuous linear functional strictly separates x and S (Theorem 3.2).

2. A SIMPLEX PROPERTY

To estimate the nonconvexity measure of an outer �-convex set, we need thefollowing simplex property, which seems to be obvious but its proof is not simple aswe have expected.

Theorem 2.1. Let S X be closed and outer �-convex. If y 2 convSnS then there existyi 2 S, i ¼ 1; 2; . . . ; k, such that y 2 convf y1; y2; . . . ; ykg and

kyi � yjk � � for all i; j 2 f1; 2; . . . ; kg: ð2:1Þ

Proof.

(a) Since z 2 convS, there are z0i 2 S, i ¼ 1; 2; . . . ; k (for some k 2 N), suchthat

z 2 Z0 :¼ convfz01; z02; . . . ; z

0kg

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(Theorem 1.24 in Valentine (1964)). Z0 is compact, because it is closed,bounded, and dimZ0

� k � 1 < 1.(b) Consider the system

F :¼ fZ : Z ¼ convfz1; z2; . . . ; zkg; zi 2 Z0\ S for 1 � i � k; and z 2 Zg

which is partially ordered by inclusion, i.e., Z0� Z00

, Z0� Z00. Since Z0 is

finite-dimensional, F is countable.(c) Let F 0 be an arbitrary totally ordered subsystem of F . Since F 0 is counta-

ble, we can number it so such that Z j� Z j 0 holds whenever Z j;Z j 0

2F 0 andj � j 0. If F 0 is finite, for instance, F 0

¼ fZ1;Z2; . . . ;Zmg, then Z � Zm for

all Z 2 F 0, i.e., Zm is an upper bound of F 0. Next, we show that F 0 also hasan upper bound in F even if it is infinite and denoted by

F 0¼ fZ j : j 2 Ng; where Z j

¼ convfz j1; z

j2; . . . ; z

jkg:

Indeed, by Tychonoff’s theorem (Dunford and Schwartz (1957), p. 32),

ðZ0\ SÞk ¼ ðZ0

\ SÞ � � � � � ðZ0\ SÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

k times

is compact in its product topology because Z0\ S is compact. Therefore,

there exists a cluster point ð �zz1; �zz2; . . . ; �zzkÞ 2 ðZ0\ SÞk of ðz j

1; zj2; . . . ; z

jkÞ,

j 2 N, which yields z 2 convf �zz1; �zz2; . . . ; �zzkg because z 2 convfz j1; z

j2; . . . ; z

jkg

for all j 2 N, i.e.,

�ZZ :¼ convf �zz1; �zz2; . . . ; �zzkg 2 F :

Assume the contrary that �zzl 62 convfzm1 ; z

m2 ; . . . ; z

mk g for some

l 2 f1; 2; . . . ; kg and some m 2 N. Then there exists a neighborhood U of�zzl such that U \ convfzm

1 ; zm2 ; . . . ; z

mk g ¼ ;, which implies

U \ convfz j1; z

j2; . . . ; z

jkg ¼ ; for j>m:

Hence, �zzl cannot be a cluster point of zjl, j 2 N, which conflicts with the

above definition. Consequently, we have �zzl 2 convfz j1; z

j2; . . . ; z

jkg for all

l 2 f1; 2; . . . ; kg and all j 2 N, which yields

�ZZ ¼ convf �zz1; �zz2; . . . ; �zzkg convfz j1; z

j2; . . . ; z

jkg ¼ Z j for all j 2 N;

i.e., �ZZ is an upper bound of F 0.(d) Since every totally ordered subsystem F 0 of F has an upper bound �ZZ 2 F , it

follows from Zorn’s lemma (Dunford and Schwartz (1957), p. 6) that thepartially ordered system F has a maximal element Y with

z2Y ¼ convfy1; y2; . . . ; ykg; yi 2 Z0\ S for 1 � i � k:

(e) Assume the contrary that Eq. (2.1) does not hold, then we can choose a pairya and yb of extreme points of Y such that kya � ybk > �. Since S is outer�-convex, there exists v 2 � ya; yb½ \S, where � ya; yb½ ¼ ½ ya; yb� n fya; ybg.

Geometrical Properties of Outer g-Convex Sets 305

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Define

Ya¼ convfya

1; ya2; . . . ; y

akg and Yb

¼ convfyb1; y

b2; . . . ; y

bkg

where

yai ¼ gf

v if yi ¼ ya

yi otherwiseand yb

i ¼ gfv if yi ¼ yb

yi otherwise:

Then we have yai ; yb

i 2 Z0\ S for 1 � i � k and Y ¼ Ya

[ Yb, whichyields z 2 Ya or z 2 Yb, i.e., Ya

2 F or Yb2 F . Since ya and yb are extreme

points of Y , it holds ya 62 Ya and yb 62 Yb, i.e., Ya and Yb are properlycontained in Y , which conflicts with the maximality of Y . Hence,kyi � yjk � � for all i; j. œ

Corollary 2.2. Let S be a closed outer �-convex set in a linear normed space. Thenx 2 convS if and only if x is contained in a finite-dimensional simplex whose verticesbelong to S and whose diameter is not greater than �.

It was observed by Brunn (1904) that, for any set S in the n-dimensional linearspace, convS ¼ Sin

if in satisfies the inequality 2in�1� n þ 1 � 2in , where Si is defined

recursively by

S1 :¼[

x; y2S

½x; y�; Siþ1 :¼[

x; y2Si

½x; y�; i � 1

(see Valentine (1964), p. 16). Applying this result to outer �-convex sets, we have

Proposition 2.3. Let S be a closed outer �-convex set in the n-dimensional linearnormed space X . Let S�

i be defined recursively as follows:

S�1 :¼

[x; y2S; kx�yk��

½x; y�; S�iþ1 :¼

[x; y2S�

i; kx�yk��

½x; y�; i � 1:

Then convS ¼ S�in

if in satisfies the inequality 2in�1� n þ 1 � 2in .

Proof. Assume y 2 convSnS. By Theorem 2.1, there exist yi 2 S, i ¼ 1; 2; . . . ; k,such that kyi � yjk � � for all i; j 2 f1; 2; . . . ; kg and y 2 convM, whereM ¼ f y1; y2; . . . ; ykg. By Brunn’s result just mentioned above, convM ¼ Min

.Since kx � yk � � for all x; y 2 M, we have Min

¼ M�in S�

in. Therefore, y 2 S�

in,

which implies convS S�in. Hence, S�

in¼ convS. œ

3. NONCONVEXITY MEASURE AND

A SEPARATION THEOREM

For S X , its relative radius and its relative center set with respect to A X isdefined by

rAðSÞ :¼ infx2A

supy2S

kx � yk and CAðSÞ :¼ fx 2 A : supy2S

kx � yk ¼ rAðSÞg: ð3:1Þ

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In particular, rX ðSÞ and rconvSðSÞ are named absolute radius and self-radius, respec-tively. Accordingly, we call

JðXÞ :¼ sup2rX ðSÞ

diamS: S X is bounded, nonempty, and nonsingleton

� �

the Jung constant and

JsðXÞ :¼ sup2rconvSðSÞ

diamS: S X is bounded, nonempty, and nonsingleton

� �

ð3:2Þ

the self-Jung constant of X , where diamS :¼ supx;y2S kx � yk is its diameter.It is worth to mention the classical result of Jung (1901) who showed

JðXÞ ¼ ð2n=ðn þ 1ÞÞ1=2 for n-dimensional Euclidean space X.

For the infinite-dimensional Hilbert space ‘2, Routledge (1952) proved Jð‘2Þ ¼ffiffiffi2

p.

Due to Klee (1960), rconvSðSÞ ¼ rX ðSÞ for all bounded sets S X if X is two-dimen-sional or an inner-product space. Therefore, JðXÞ and JsðXÞ are coincided in suchspaces. Hence, we have (see Maluta (1984))

Jsð‘n2Þ ¼ ð2 n=ðn þ 1ÞÞ1=2 and Jsð‘2Þ ¼

ffiffiffi2

p: ð3:3Þ

Amir (1985) showed the upper bound

JsðXÞ �2n

n þ 1if dimX ¼ n: ð3:4Þ

It follows form rX ðSÞ � rconvSðSÞ and the result of Bohnenblust Bohnenblust(1938), Grunbaum (1959), and Leichtweiss (1955), that there is no smaller upperbound which holds true for all n-dimensional normed spaces.

Due to Pichugov (1988),

Jsð‘pÞ ¼ JsðLp½0; 1�Þ ¼ maxf21=p; 21�1=pg; 1 � p < 1: ð3:5Þ

Self-Jung constant for further spaces can be found in Maluta (0000), for instance

JsðXÞ ¼ 1 if X ¼ ðR2; k � k1Þ;

JsðXÞ ¼ 2 if X is a nonreflexive Banach space.ð3:6Þ

Now we can use the self-Jung constant or its upper bounds, as given by Eqs.(3.3)–(3.6), to estimate the nonconvexity measure of an outer �-convex set inconcrete normed spaces.

Theorem 3.1. Suppose S X is closed and outer �-convex. Then

8y 2 convSnS 9z 2 S : ky � zk �1

2JsðXÞ �;

i.e., convS S þ �BBð0; rÞ if r � ð1=2Þ JsðXÞ �, where �BBð y; rÞ ¼ fx 2 X : kx � yk � rg.


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Proof. By Theorem 2.1, for y 2 convSnS, there exists Sk ¼ f y1; y2; . . . ; ykg S suchthat diamSk � � and y 2 convSk. Without loss of generality, assume thaty1; y2; . . . ; yk are linearly independent and y 2 riðconvSkÞ, where riA denotes therelative interior of A. Let us prove by induction that

min1�i�k

ky � yik �1

2JsðXÞ �: ð3:7Þ

For k ¼ 2, JsðXÞ � 1 yields

minfky � y1k; ky � y2kg �1

2ky1 � y2k �

1

2JsðXÞ �:

Assume that Eq. (3.7) is true for 2 � k � l, we show now that it holds fork ¼ l þ 1, too. Take a c from the relative center set CconvSk

ðSkÞ which is nonempty.By Eqs. (3.1)–(3.2), we have

max1�i�k

kc � yik � rconvSkðSkÞ �

1

2JsðXÞ diamSk �

1

2JsðXÞ �: ð3:8Þ

If y ¼ c then Eq. (3.7) follows from Eq. (3.8). Otherwise, consider the ray from cthrough y which cuts the boundary convSknriðconvSkÞ at some point

y0 2 convSk0 ; where Sk0 ¼ fyi1 ; yi2 ; . . . ; yik0g Sk

and k0� k � 1 ¼ l. If y0 2 Sk then y 2 ½c; y0� and Eqs. (3.1)–(3.2) imply

ky � y0k � kc � y0k � rconvSkðSkÞ �

1

2JsðXÞ �:

If y0 62 Sk then we choose Sk0 so that y0 2 riðconvSk0 Þ. By induction assumption, thereexits z0 2 Sk0 S such that ky0 � z0k � ð1=2Þ JsðXÞ �. This and Eq. (3.8) yield finally

ky � z0k � maxfkc � z0k; ky0 � z0kg �1

2JsðXÞ �:

Hence, Eq. (3.7) is always true. The rest is obvious. œ

Theorem 3.1 says that the nonconvexity measure of a closed outer �-convexsubset of some normed linear space X is not greater than ð1=2Þ JsðXÞ �. By usingthis result, we can derive the following separation theorem, where S is not necessarilyclosed.

Theorem 3.2. Suppose S X is outer �-convex. If �BBðx; rÞ \ S ¼ ; for somer > ð1=2Þ JsðXÞ � then x 62 clðconvSÞ, and therefore, some nonzero continuous linearfunctional strictly separates x and S.

Proof. By definition and Theorem 1.27 in Valentine (1964), we have

clðconvSÞ ¼ convS ¼ convðclSÞ ¼ clðconvðclSÞÞ;

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where convA denotes the closed convex hull of A. Therefore, if x 2 clðconvSÞ thenthere exists a y 2 convðclSÞ such that kx � yk < =2, where ¼ r � ð1=2Þ�JsðXÞ � > 0. By Proposition 2.4 in Phu and An (1999), clS is outer �-convex, too.Consequently, due to Theorem 3.1, there exists z 2 clS such that ky � zk �

ð1=2Þ JsðXÞ �. For any z0 2 S satisfying kz � z0k < =2 we have

kx � z0k � kx � yk þ ky � zk þ kz � z0k <1

2JsðXÞ � þ ¼ r;

which conflicts with �BBðx; rÞ \ S ¼ ;. Hence, x 62 clðconvSÞ. The rest follows fromclassical separation theorem (see e.g., Dunford and Schwartz (1957), p. 417). œ

Actually, x 62 convS follows from �BBðx; rÞ \ S ¼ ; even for r ¼ ð1=2Þ JsðXÞ �.Therefore, if in addition X is finite-dimensional and S is compact then some non-zero continuous linear functional strictly separates x and S, because convS iscompact (see Valentine (1964), p. 40).

REFERENCES

Amir, D. (1985). On Jung’s constant and related constants in normed linear spaces.Pacific J. Math. 118:1–15.

Bohnenblust, F. (1938). Convex regions and projections in Minkowski spaces. Ann.of Math. 39:301–308.

Brunn, H. (1904). Uber das Durch eine Beliebige Endliche Figur Bestimmte Eigebilde.Leipzig: Boltzmann-Festschrift, 94–104.

Dunford, N., Schwartz, J. T. (1957). Linear Operators—Part I: General Theory.New York: Interscience Publishers, Inc.

Grunbaum, B. (1959). On some covering and intersection properties in Minkowskispaces. Pacific J. Math. 9:487–494.

Jung, H. E. W. (1901). Uber die kleinste Kugel, die eine raumliche Figur einschließt.J. Reine Angew. Math. 123:241–257.

Klee, V. (1960). Circumspheres and inner products. Math. Scand. 8:363–370.Leichtweiss, K. (1955). Zwei extremalprobleme der Minkowski-geometrie. Math. Z.

62:37–49.Maluta, E. (1984). Uniformly normal structure and related coefficients. Pacific J.

Math. 111:357–369.Phu, H. X., An, P. T. (1999). Outer �-convexity in normed linear spaces. Vietnam J.

Math. 27:323–334.Pichugov, S. A. (1988). On Jung’s constant of the space Lp. Mat. Zemetki

43:604–614, English transl.: Math. Notes 43:348–354.Routledge, N. (1952). A result in Hilbert space. Quart. J. Math. 3:12–18.Valentine, F. A. (1964). Convex Sets. New York: McGraw-Hill Book Company.


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Some Geometrical Properties of Outer γ-Convex Sets

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