Background on Convex Sets

Background on convex sets The Brunn-Minkowski theory is the classical core of the geometry of convex bodies.. In recent decades, the theory of convex bodies has expanded considerably; ne

Background on convex sets

The Brunn-Minkowski theory is the classical core of the geometry of convex bodies It originated with the thesis of Hermann Brunn in 1887 and is

in its essential parts the creation of Hermann Minkowski, around the turn of the century

In recent decades, the theory of convex bodies has expanded considerably; new topics have been developed and originally neglected branches of the subject have gained in interest

Nevertheless, the Brunn-Minkowski theory has remained of constant interest owing to its various new applications, its connections with other fields, and the challenge of some resistant open problems

Aiming at a brief characterization of Brunn-Minkowski theory, one might say that it is the result of merging two elementary notions for point sets in Euclidean space: vector addition and volume The vector addition of convex bodies, usually called Minkowski addition, has many facets of independent geometric interest Combined with volume, it leads to the fundamental Brunn-Minkowski inequality and the notion of mixed volumes The latter satisfy a series of inequalities which, due to their flexibility, solve many extremal problems and yield several uniqueness results Looking at mixed volumes from a local point of view, one is led to mixed area measures

Quermassintegrals, or Minkowski functionals, and their local versions, surface area measures and curvature measures, are a special case of mixed volumes and mixed area measures They are related to the differential geometry of convex hypersurfaces and to integral geometry

The basic properties of convex bodies and thus lays the foundations for subsequent developments, does not claim much originality; in large parts, it follows the procedures in standard such as McMullen & Shephard [5], Roberts & Varberg [6], and Rockafellar [7] It serves as a general introduction to the metric geometry of convex bodies

The recognition of the subject of convex functions as one that deserves to

be studied in its own is generally ascribed to J L W V Jensen [8], [9]

During the whole 20th Century an intense research activity was done and cant results were obtained in geometric functional analysis, mathematical economics, convex analysis, nonlinear optimization etc A great role in the popularization of the subject of convex functions was played by the famous book of G H Hardy, J E Littlewood and G Pólya [10], on inequalities Roughly speaking, there are two basic properties of convex functions that made them so widely used in theoretical and applied mathematics: The maximum is attained at a boundary point and any local minimum is a global one, moreover a strictly convex function admits at most one minimum

Here we shall fix our notation and collect some basic definitions and at the core of the notion of convexity is the comparison of means We shall work in n-dimensional real Euclidean vector space n

E with origin 0, scalar product .,. and induced norm . We shall not distinguish formally between the vector space n

E and its corresponding affine space

The vector n

xE is a linear combination of the vectors x x1, 2, ,x kE n if

x x   x    x with suitable   1 , 2 , , k If such i , exist with

       then x is an affine combination of x x1, 2, ,x k For n

AE ,

linA(affA) denotes the linear hull (affine hull) of A this is the set of all linear (affine) combinations of elements of A and at the same time the smallest linear subspace (affine subspace) of n

E containing A Points x x1, 2, ,x kE n

are affinely independent if none of them is an affine combination of the others, i.e., if

1

k

i i i

x o







 with i and

1

0

k i i









implies that  1 2   k  0 This is equivalent to the linear independence

of the vectors x2x1, ,x k x1

We may also define a map  :E nE n by    x  x,1 then

1 , 2 , , k n

x x x E are affinely independent if and only if    x1 ,  x2 , ,  x k are linearly independent

By clA,int ,A bdA we denote, respectively, the closure, interior and boundary of a subset A of a topological space For n

AE , the sets relint A, relbdA are the relative interior and relative boundary, that is the interior and boundary of A relative to its affine hull

The scalar product in n

E will often be used to describe hyperplanes and half spaces A hyperplane of n

E can be written in the form

u

H   xE x u  

with uE n \ o and   ; here H u, H v, if and only if v,    u,  with

0

  We say that u is a normal vector of H u,

We also use .,. to denote the scalar product on n

E  give by

  x,  , y,   x y,  

1 Basic convexity

a Convex sets

A set AE is convex if together with any two points x y, it contains the segment  x y, , thus if

1  x yA for x y, A;   0,1

As immediate consequences of the definition note that intersections of convex sets are convex, affine images and pre-images of convex sets are convex and if A B, are convex then AB and A,   are convex

We know, for AE n,   0,   0 one trivially has A A   A Equality (for all   ,  0) holds precisely if A is convex In fact, if A is convex and x A A then x a b with a b, A and hence

   

  thus A A   A

A set n

AE is called a convex cone if A is convex and nonempty and if

, 0

xA  implies xA Thus a nonempty set n

AE is a convex cone if and only if A is closed under addition and under multiplication by non-negative real numbers

The point n

xE is a convex combination of the points x x1, 2, ,x nE n if there are numbers   1 , 2 , , k such that

The vector n

xE is a positive combination of the vectors x x1, 2, ,x kE n if

1

k

i i i

x x



 with i  0, i 1,k

AE the set of all convex combinations (positive combinations) of any finitely many elements of A is called the convex hull (positive hull) of A

and is denoted by conv A (pos A) We have theorem,

Theorem 1.1 If n

AE is convex, then convAA For an arbitrary set n

AE ,

convAis the intersection of all convex subsets of n

E containing A If

A BE , then conv A B  convA convB

Proof: We can see ([2], p.2-3) An immediate consequence is that

 

conv convA convA As above Theorem 1.1, we have

Theorem 1.2 If n

AE is a convex cone, then posAA For a nonempty set

n

AE , posA is the intersection of all convex cones in n

E containing A If

A BE , then pos A B   posA posB

Theorem 1.3 (Caratheodory's theorem) If n

AE and xconvA, then x is a convex combination of affinely independent points of A In particular, x is a convex combination of n 1 or fewer points of A

Proving the theorem we can see ([2], p.3) The convex hull of finitely many points is called a poly tope A k-simplex is the convex hull of k + 1 affinely independent points and these points are the vertices of the simplex Thus Caratheodory's theorem states that convA is the union of all simplices with vertices in A

Theorem 1.4 (Radon's theorem) Each set of affinely dependent point (in

particular, each set of at least n 2 points) in n

E can be expressed as the union of two disjoint sets whose convex hulls have a common point

Proof If x x1, 2, ,x k are af finely dependent, there are numbers

1 , 2 , , k

    , not all zero, with

1

0

k

i i i x







1

0

k i i







We may assume after renumbering, that i  0precisely for i 1, 2, ,j; then 1  j k with

we obtain

i i j

and thus xconv x x 1 , 2 , ,x jconv x j1 ,x j2 , ,x k The assertion follows

Theorem 1.5 (Helly's theorem) Let A A1, 2, ,A k E n be convex sets If any

1

n of these sets have a common point, then all the sets have a common point

Theorem 1.6 Let M be a finite family of convex sets in n

E and let n

KE

be convex If any n 1 elements of M are intersected by some translate of K, then all elements of M are intersected by a translate of K

Lemma 1.1 Let n

AE be convex If x intA and yclA, then x y,  intA

Theorem 1.7 If n

AE is convex, then int A and clA are convex If n

AE is open, then convA is open

Proof All Theorem 1.5; 1.6; 1.7 and Lemma 1.1, we see ([2], p.4-5)

b The metric projection

n

AE is a fixed nonempty closed convex set To each n

xE there exists a unique point p A x , A satisfying

 , 

xp A x  x y for all yA

In fact, for suitable  the set B x , A compact and nonempty, hence the continuous function y xy attains a minimum on this set, say at y0, then

xy  x y  y A

If y1 A satisfies xy0  x y,  y A , then 0 1

2

y y

0

x  z x y , exept if y0  y1 Thus y0  p A x , ) is unique

In this way a map p A , :E n A is defined; it is called the metric projection or nearest-point map of A It will play an essential role in Chapter 4 ([2], p.197-269) when the volume of local parallel sets is investigated It also provides a simple approach to the basic support and separation properties of convex sets (see the next section), as used by Botts (1942) and McMullen & Shephard [4]

We have xp A x ,  d A x ,  and for xE n\A we denote by

 ,    , 

,

x p A x

u A x

d A x





the unit vector pointing from the nearest point p A x ,  to x and by

 ,    ,   ,  0

R A x  p A x  u A x  

the ray through x with endpoint p A x , 

Lemma 1.2 Let xE n\A and yR A x , , then p A x ,  p A y , 

Proof Suppose that p A x ,  p A y ,  If y x p A x,  ,  then

 

 

, ,

x p A x x y y p A y

x y y p A x

x p A x y

which is a contradiction If x y p A x,  ,   , let q p A x ,  ,p A y,  be the point such that the segment  x q, is paraller to y p A y,  ,  Then

     

, 1

y p A y

x q

x p A x y p A x





again a contradiction

Theorem 1.8 The metric projection is contracting, that is,

 ,   , 

p A x p A y  x y for x y, E n

Theorem 1.9 Let S be a sphere containing A in its interior Then

 , 

p A S bdA

Proof Theorem 1.9 and Theorem 1.10, we can see ([2],p.10)

The existence of a unique nearest-point map is characteristic of convex sets We prove this result here to complete the picture, although no use will

be made of it

Theorem 1.10 Let n

AE be a closed set with the property that to each point

of n

E there is a unique nearest point in A Then A is convex

Proof Suppose A satisfies the assumption but is not convex Then there are points x y, with  x y,  A  x y, and one can choose p 0 such that the

2

x y

B B  

By an elementary compactness argument, the family B of all closed balls containing '

B and satisfying  '

int B  A contains a ball C with maximal radius

By this maximality, there is a point p C A and by the assumed uniqueness of nearest points in A it is unique If bd B and bdC have a common point, let this (unique) point be q, otherwise let q be the centre of

B For sufficiently small   0, the ball C qp includes B and does not meet A

Hence, the family B contains an element with greater radius than that of

C, a contradiction

c Convex functions

For convex functions it is convenient to admit as the range the extended system     ,  of real numbers with the usual rules These are the following conventions

For   ,     , ,                   ,

0,  0

For a given function f E: n  and for   we use the abbreviation

   n   

f    xE f x   , and f   , f   etc are defined similarly

A function f E: n  is called convex if f is proper, which mean that

f     and f    , and if

(1 )  1     

f   x y    f x  f y

for all x y, E n and for 0    1 A function f D:  with n

DE is called convex if its extension f defined by

 

\

n

f x for x D f

for x E D



 



is convex A function f is concave if f is convex

Trivial examples of convex functions are the affine functions; these are the functions f E: n  of the form f x  u x,   with uE n,   A real-valued function on n

E is affine if and only if it is convex and concave

The following assertions are immediate consequences of the definition The supremum of (arbitrarily many) convex functions is convex if it is proper

If f g, are convex functions, then f g and f for   0 are convex if they are proper

Remark 1.1 If f is convex, then

 1 1 2 2 k k 1  1 2  2 k  k

f x   x    x   f x   f x    f x

for all x x1, 2, ,x kE n and   1 , 2 , , k 0,1 with   1  2   k  1 This is called Jensen's inequality; it follows by induction

Convex functions have the important property (important for optimization etc.) that each local minimum is a global minimum

In fact, let f E: n  be convex and suppose that x0E n,   0 are such that f x 0   and f x 0  f x  for xx0   For n

xE with xx0   let

0

1

then yx0   and hence

 0      0

1

which gives f x 0  f x 

A convex function determines in a natural way several convex sets Let

: n

f E  be convex Then the sets

 

domf  f  

the effective domain of f and for   the sublevel sets f   , f   are convex The epigraph of

   

epif  x E  f x  

is a convex subset of n

E  The asserted convexity is in each case easy to see Vice versa, a nonempty convex set n

AE determines a convex function

by

  0

\

for x A

I x

for x E A





 



the indicator function of A

Theorem 1.11 Each convex function f E: n  is continuous on int dom f

and Lipschitzian on any compact subset of int dom f

Theorem 1.12 Let : n

f E  be convex Then on intdom f the following holds The right derivative '

r

f and the left derivative '

r

f exist and are monotonically increasing functions The inequality ' '

f  f is valid and with the exception of at most countably many points, ' '

f  f holds and hence f is differentiate Further, '

r

f is continuous from the right and '

1

f is continuous from the left (in particular, if f is differentiable on intdom f , then it is continuously differentiable)

Proof We can see [2], page 23-25

Remark 1.2 Let f :  be convex, let x0 and m be a number with

   

f x  m f x As noted in the above proof, one has

     

 

'

0

'

r

f x m if x x

f x f x



thus f x  f x 0 m x x0 for all x This shows that the line

 x y,   y f x0 m xx0 

supports the epigraph of f at the point x0 ,f x 0 

2 Background on convex

a The Hahn-Banach extension theorem

Throughout, E will denote a real linear space

A functional p A: E is subadditive if p x y p x    p y  for all

,

x yE; pis positively homogeneous if p x  p x  for each   0and each

x in E ; p is sublinear if it has both the above properties A sublinear functional p is a seminorm if p x   p x  for all scalars Finally, a seminorm pis a norm if

  0 0

p x   x

If p is a sublinear functional, then p 0  0 and  p   x p x. If p is a seminorm, then p x  0 for all xE and x p x  0is a linear subspace of

E

Theorem 2.1 (The Hahn-Banach theorem) Let pbe a sublinear functional on

E, let E0 be a linear subspace of E and let f0 :E0  be a linear functional dominated by p , that is f0   x p x. for all xE0 Then f0 has a linear extension f to E which is also dominated by p

Proof We consider the set P of all pairs h H, , where H is a linear subspace of E that contains E0 and h H:  is a linear functional dominated

by p that extends f0.P is nonempty (as f E0 , 0P) One can easily prove that P is inductively ordered with respect to the order relation

   ' ' '

h H h H H H and '

h H h,

so that by Zorn's lemma we infer that P contains a maximal element g G, 

It remains to prove that GE

If GE, then we can choose an element zE G\ and denote by '

G the set of all elements of the form x z, with xG and   Clearly, G' is a linear space that contains G strictly and the formula

   

'

g x z g x  

de_nes (for every   ) a linear functional on '

G that extends g We shall show that  can be chosen so that '

g is dominated by p (a fact that contradicts the maximality of g G, )

In fact, '

g is dominated by pif

  .  

g x   p x z for every xG and every  

If   0this means:

  .  

g x   p xz for every xG

If   0, we get g x   p x z for every xG

Therefore, we have to choose  such that

    .    

g u p xz  p v z g v for every u v G,  This choice is possible because

      .  .   

g u g v g uv p uv p u z p vz

for every u v G,  , which yields

v G

u G

g u p u z p v z g v



Corollary 2.1 If p is a sublinear functional on a real linear space E, then for every element x0 E there exists a linear functional f E:  such that

 0  0

f x  p x and f x p x   . for all xE

Proof Take E0  x0   and f0x0 p x 0 in Theorem 2.1

The continuity of a linear functional on a topological linear space means that it is bounded in a neighborhood of the origin

In the case of normed linear spaces E, this makes it possible the norm of a continuous linear functional f E:  by the formula

 

.1

sup

x

f  f x

We shall denote by '

E the dual space of E that is, the space of all continuous linear functionals on E, endowed with the norm above The dual space is always complete (every Cauchy sequence in E is also converging)

It is worth to notice the following variant of Theorem 2.1 in the context of real normed linear spaces:

Theorem 2.2 (The Hahn-Banach theorem) Let E0 be a linear subspace of the normed linear space E , and let f0:E0  be a continuous linear functional Then f0 has a continuous linear extension f to E, with f  f0

Corollary 2 2: If E is a normed linear space, then for each x0 E with x0  0

there exists a continuous linear functional f E:  such that f x 0  x0 and

1

f 

Corollary 2.3: If E is a normed linear space and x is an element of E such that f x  0 for all f in the dual space of E, then x 0

The weak topology on Eis the locally convex topology associated to the family of seminorms

  sup  : 

F

p x  f x f F , where F runs over all nonempty subsets of '

E A sequence  x n n converges

to x in the weak topology (abbreviated, w

n

x  x ) if and only if

   

: n

f x  f x for every '

f E When n

E this is the coordinate-wise convergence and agrees with the norm convergence In general, the norm function is only weakly lower semicontinuous that is,

.lim inf

w

n



By Corollary 2.3 it follows that '

E separates E in the sense that

,

x yE and f x  f y  for all '

f E  x y