Background on Convex Sets

Background on convex sets Full name: LAI DUC NAM 姓名：赖德南 Student ID: 11860027 学号：11860027 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 20 th 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 xE is a linear combination of the vectors 12 , , , n k x x x E if 1 1 2 2 kk x x x x        with suitable 12 , , , k     . If such i  , exist with 12 1 k        then x is an affine combination of 12 , , , k x x x . For n AE , 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 12 , , , n k x x x E are affinely independent if none of them is an affine combination of the others, i.e., if 1 k ii i xo     with i   and 1 0 k i i     implies that 12 0 k        . This is equivalent to the linear independence of the vectors 2 1 1 , , k x x x x . We may also define a map : nn EE   by     ,1xx   then 12 , , , n k x x x E are affinely independent if and only if       12 , , , k x x x    are linearly independent. By ,int ,clA A bdA we denote, respectively, the closure, interior and boundary of a subset A of a topological space. For n AE , the sets relint A , relbd A 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   , , n u H x E x u      with   \ n u E o and   ; here ,,uv HH   if and only if     ,,vu     with 0   . We say that u is a normal vector of ,u H  . 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 AE is convex if together with any two points ,xy it contains the segment   ,xy , thus if   1 x y A     for   , ; 0,1x y A   . 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 ,AB are convex then AB and   ,A   are convex. We know, for , 0, 0 n AE     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     x a b A                   thus   A A A        . A set n AE is called a convex cone if A is convex and nonempty and if ,0xA   implies xA   . Thus a nonempty set n AE is a convex cone if and only if A is closed under addition and under multiplication by non- negative real numbers. The point n xE is a convex combination of the points 12 , , , n n x x x E if there are numbers 12 , , , k     such that   11 , 0, 1, , 1 kk i i i i ii x x i k          . The vector n xE is a positive combination of the vectors 12 , , , n k x x x E if 1 k ii i xx     with   0, 1, i ik   . For n AE 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 AE is convex, then convA A . For an arbitrary set n AE , convA is the intersection of all convex subsets of n E containing A . If , n 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 AE is a convex cone, then posA A . For a nonempty set n AE , posA is the intersection of all convex cones in n E containing A . If , n A B E , then   pos A B posA posB   . Theorem 1.3 (Caratheodory's theorem). If n AE and x convA , then x is a convex combination of affinely independent points of A . In particular, x is a convex combination of 1n 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 2n points) in n E can be expressed as the union of two disjoint sets whose convex hulls have a common point. Proof. If 12 , , , k x x x are af finely dependent, there are numbers 12 , , , k     , not all zero, with 1 0 k ii i x     and 1 0 k i i     . We may assume after renumbering, that 0 i   precisely for 1,2, ,ij ; then 1 jk with   1 2 1 2 0 j j j k                   we obtain 11 j k ii ii i i j x x x           and thus     1 2 1 2 , , , , , , j j j k x conv x x x conv x x x   . The assertion follows. Theorem 1.5 (Helly's theorem). Let 12 , , , n k A A A E be convex sets. If any 1n 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 KE be convex. If any 1n 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 AE be convex. If intxA and y clA , then   , intx y A . Theorem 1.7. If n AE is convex, then int A and clA are convex. If n AE 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 AE is a fixed nonempty closed convex set. To each n xE there exists a unique point   ,p A x A satisfying   ,x p A x x y   for all yA . 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 0 y , then 0 ,x y x y y A     . If 1 yA satisfies 0 ,x y x y y A     , then 01 2 yy zA   and 0 x z x y   , exept if 01 yy . Thus   0 ,y p A x ) is unique. In this way a map   ,. : n p A E 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 \ n x E 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         , , , 0R A x p A x u A x     the ray through x with endpoint   ,p A x . Lemma 1.2. Let \ n x E 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   ,xq is paraller to   ,,y p A y   . Then       , 1 ,, y p A y xq 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 , n x y E . 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 AE 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 ,xy with     ,,x y A x y and one can choose 0p  such that the ball , 2 xy BB       . By an elementary compactness argument, the family B of all closed balls containing ' B and satisfying   ' int BA   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 bd C 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   ,,      ,                  ,         and finally    , 0 or  according to whether 0, 0     or 0   . For a given function : n fE and for   we use the abbreviation       n f x E f x      , and     ,ff   etc. are defined similarly. A function : n fE is called convex if f is proper, which mean that   f     and   f     , and if         (1 ) 1f x y f x f y          for all , n x y E and for 01   . A function :fD with n DE 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 : n fE of the form   ,f x u x   with , n uE   . 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 ,fg are convex functions, then fg and f  for 0   are convex if they are proper. Remark 1.1. If f is convex, then         1 1 2 2 1 1 2 2 k k k k f x x x f x f x f x              for all 12 , , , n k x x x E and   12 , , , 0,1 k     with 12 1 k        . 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 : n fE be convex and suppose that 0 ,0 n xE   are such that   0 fx  and     0 f x f x for 0 xx   . For n xE with 0 xx   let 0 00 1y x x x x x x          , then 0 yx   and hence         00 00 1f x f y f x f x x x x x           which gives     0 f x f x . A convex function determines in a natural way several convex sets. Let : n fE be convex. Then the sets   domf f   the effective domain of f and for   the sublevel sets     ,ff   are convex. The epigraph of       , n 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 AE determines a convex function by   0 \ A n for x A Ix for x E A       the indicator function of A . Theorem 1.11. Each convex function : n fE is continuous on int dom f and Lipschitzian on any compact subset of int dom f . Theorem 1.12. Let : n fE 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 '' 1 r ff is valid and with the exception of at most countably many points, '' 1 r ff 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 0 x  and m be a number with     '' 1 0 0r f x m f x . As noted in the above proof, one has         ' 0 0, 0 ' 0 1 0 0 , r f x m if x x f x f x xx f x m if x x              thus       00 f x f x m x x   for all x . This shows that the line         00 ,x y y f x m x x     supports the epigraph of f at the point     00 ,x f x . 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 ; p is positively homogeneous if     p x p x   for each 0   and 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 p is a norm if   00p x x   . If p is a sublinear functional, then   00p  and     .p x p x . If p is a seminorm, then   0px for all xE and     0x p x  is a linear subspace of E . Theorem 2.1. (The Hahn-Banach theorem). Let p be a sublinear functional on E , let 0 E be a linear subspace of E and let 00 :fE be a linear functional dominated by p , that is     0 .f x p x for all 0 xE . Then 0 f has a linear extension f to E which is also dominated by p . Proof. We consider the set P of all pairs   ,hH , where H is a linear subspace of E that contains 0 E and :hH is a linear functional dominated by p that extends 0 f . P is nonempty (as   00 ,f E 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   ,gG . It remains to prove that GE . If GE , then we can choose an element \z E G and denote by ' G the set of all elements of the form xz   , with xG 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   ,gG ). In fact, ' g is dominated by p if     .g x p x z    for every xG and every   . If 0   this means:     .g x p x z   for every xG . If 0   , we get     .g x p x z   for every xG . 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     sup ( ) ( ) .inf ( ) ( ) vG uG 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 0 xE there exists a linear functional :fE such that     00 f x p x and     .f x p x for all xE . Proof. Take   00 Ex   and     0 0 0 f x p x   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 :fE 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 0 E be a linear subspace of the normed linear space E , and let 00 :fE be a continuous linear functional. Then 0 f has a continuous linear extension f to E , with 0 ff . Corollary 2 .2: If E is a normed linear space, then for each 0 xE with 0 0x  there exists a continuous linear functional :fE such that   00 f x x and 1f  . Corollary 2.3: If E is a normed linear space and x is an element of E such that   0fx for all f in the dual space of E , then 0x  . The weak topology on E is 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   n n x converges to x in the weak topology (abbreviated, w n xx ) if and only if     : n f x f x for every ' fE . 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, .liminf w nn n x x x x    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   . [...]... smallest closed convex set coA containing A (that is, the intersection of all closed convex sets containing A ) From Theorem 6 we can infer the following result on the support of closed convex sets: Corollary 2.5 If A is a nonempty subset of a real locally convex Hausdorff space E , then the closed convex hull co  A is the intersection of all the closed half-spaces containing A Equivalently, co ... Niculescu, Lars-Erik Persson, Convex functions and their applications a contemporary approach, page 103-174, 197-206 [4] McMullen, P & Shephard G C 1971, Convex Poly topes and the Upper Bound Conjecture, Cambridge Univ Press [5] McMullen, P & Shephard G C 1971, Convex Poly topes and the Upper Bound Conjecture, Cambridge Univ Press [6] Roberts, A W & Varberg, D E 1973, Convex Functions, Academic Press, New... A       x f  x  sup f  y   f E '  yA    Corollary 2.6 In a real locally convex Hausdorff space E , the closed convex sets and the weakly closed convex sets are the same Finally it is worth to mention a non-topological version of the separation results above, which is important in optimization theory Given a set A in a linear space E , a point a of A is said to be a core point if... open convex neighborhood W of the origin such that  K1  W  \  K 2  W    This follows by using reduction absurdum Since the sets K1  W and K 2  W are convex and open, from Lemma 2.1 we infer the existence of a separating hyperplane H A moment's re¡ection shows that H separates ã strictly K1 from K 2 The closed convex hull of a subset A of a locally convex space E is the smallest closed convex. .. the discussion above shows that the closed hyperplanes H  E coincide with the constancy sets of nonzero continuous and linear functionals In fact, it suffces to consider the case where H is a closed subspace of codimension 1 In that case E / H is 1-dimensional and thus it is algebraically and topologically isomorphic to By composing such an isomorphism with the canonical projection from E onto E / H... onto E / H we obtain a continuous linear functional h for which H  ker h To each hyperplane  x h( x)   we can attach two half-spaces,  x h( x). and x h( x)   We say that two sets A, B are separated by the hyperplane H if they are contained in different half-spaces The separation is strict if at least one of the two sets does not intersect H c Separation of Convex Sets Theorem 2.5 (Mazur's... (Strong separation theorem) Let K1 and K 2 be two nonempty convex sets in a real locally convex Hausdorff space E such that K1 \ K 2   If K1 is compact and K 2 is closed then there exists a closed hyperplane strictly separating K1 from K 2 Particularly, if K is a closed convex set in a locally convex space E and x  E is not in K then there exists a functional f  E ' such that f  x   sup  f  x... 1970, Convex Analysis, Princeton Univ Press, Princeton, NJ [8] J L W V Jensen, Om konvexe Funktioner og Uligheder mellem Middelvaerdier, Nyt Tidsskr Math., 16B (1905), 49-69 [9] J L W V Jensen, Sur les fonctions convexes et les inegalites entre les valeurs moyennes, Acta Math., 30 (1906), 175-193 [10] G H Hardy, J E Littlewood and G Pó lya, Inequalities, Cambridge Mathematical Library, 2nd Edition, 1952,... Since K1 and K 2 are convex, U is convex too Moreover, 0 U since K1 \ K 2   By Corollary 2.4 there exists a continuous linear functional f such that f  x   0 on U Therefore f  x   f  y  , x  K1 , y  K 2 Letting   inf  f  x  x  K1 , one can show immediately that K1 and K 2 are separated by the closed hyperplane H  x f  x     Theorem 2.6 (Strong separation theorem) Let K1... every  with    Theorem 2.7 Let K and M be two nonempty convex sets in a real linear space E If K contains core points and M contains no core point of K then K and M can be separated by a hyperplane References [1] Elliott H.Lieb, Michael Loss, Analysis second edition, Graduate Studies in Mathematics, Vol.14 [2] R.S Doran, J.Goldman, T.Y.Lam, E Lutwak, Convex bodies: the BrunnMinkowski theory, Cambrideg . A   . 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 ,AB are convex then AB and. 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 ,fg are convex. intersection of all convex subsets of n E containing A . If , n 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 .