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Basic (And Some More) Facts from Convexity

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Basic (And Some More) Facts from Convexity 1 Basic (and some more) facts from Convexity In this chapter we will briefly collect some basic results and concepts con- cerning convex bodies, which may be considered as the main ”continuous” component/ingredient of the Geometry of Numbers. For proofs of the results we will refer to the literature, mainly to the books [2, 5, 6, 7, 9, 11, 13, 14], which are also excellent sources for further information on the state of the art concerning convex bodies, polytopes and their applications. As usual we will start with setting up some basic notation; more specific ones will be introduced when needed. n Let R = x =(x1,...,xn)| : xi R be the n-dimensional Euclidean { 2 } space, equipped with the standard inner product x, y = n x y , x, y h i i=1 i i 2 Rn. P For a subset X Rn,letlinX and a↵ X be its linear and affine hull, ✓ respectively, i.e., with respect to inclusions, the smallest linear or affine subspace containing X.IfX = we set lin X = 0 , and a↵ X = .The ; { } ; dimension of a set X is the dimension of its affine hull and will be denoted by dim X.ByX +Y we mean the vectorwise addition of subsets X, Y Rn, ✓ i.e., X + Y = x + y : x X, y Y . { 2 2 } The multiplication λX of a set X Rn with λ R is also declared vec- ✓ 2 torwise, i.e., λX = λ x : x X .Wewrite X =( 1) X, X Y = { 2 } − − − X +( 1) Y and x + Y instead of x + Y . − n { } AsubsetC R is called convex, i↵for all c1, c2 C and for all ✓ 2 λ [0, 1] hold λ c +(1 λ) c C. A convex compact (i.e., bounded and 2 1 − 2 2 closed) subset K Rn is called a convex body. The set of all convex bodies ⇢ in Rn is denoted by n. K n is called o-symmetric,i.e.,symmetricwith K 2K respect to the origin, if K = K. The set of all o-symmetric convex bodies − in Rn is denoted by n. Ko There is a simple one-to-one correspondence between n-dimensional o- n n symmetric convex bodies and norms on R : given a norm : R R 0 |·| ! ≥ 1 2 1. Basic (and some more) facts from Convexity we consider its unit ball x Rn : x 1 ,whichisao-symmetric convex { 2 | | } body by the definition of a norm. For the reverse correspondence we need the notion of a distance function. For K n,dimK = n, the function 2Ko n K : R R 0 given by x K := min ⇢ R 0 : x ⇢K |·| ! ≥ | | { 2 ≥ 2 } is called the distance function of K. By the convexity and symmetry of K it is easily verified that is |·|K indeed a norm. A well-studied family of o-symmetric convex bodies are the p n unit balls Bn := x R : x 1 associated to the p-norms { 2 | |p } n 1/p p x := xi ,p [1, ), and x := max xi . | |p | | 2 1 | | 1 i n{| |} ! 1 Xi=1 In the Euclidean case, i.e., p = 2, we denote the Euclidean norm just by |·| and its unit ball by Bn. 0 1 1 1 −1 1 1 1 − 0 0 0 0 0 0 0 1 1 1 −1 1 − − − p=1 p =2 p =4 p = 1 Figure 1.1: Unit balls of p-norms n Bn1 = x R : 1 xi 1 is the o-symmetric (regular) cube with { 2 −1 n} n edge length 2, and B = x R : xi 1 is called the (regular) n { 2 i=1 | | } crosspolytope. P One big advantage of convex sets K is their property that points not belonging to K can easily be separated from K. In order to formulate this precisely and a bit more generally, we denote for a Rn 0 , b R 2 \{ } 2 by H(a,b)= x Rn : a, x = b the hyperplane with normal vector a { 2 h i } and right hand side b. The associated two closed halfspaces are denoted by H (a,b) and H (a,b), respectively, i.e., H (a,b)= x Rn : a,x b ≥ { 2 h i } and H (a,b)=H ( a, b). ≥ − − n 1.1 Theorem [Separation Theorem]. Let C1,C2 R be convex with ⇢ C C = . 1 \ 2 ; i) Then there exists a separating hyperplane H(a,b),i.e.,C1 H (a,b) ✓ and C2 H (a,b). ✓ ≥ ii) If C is compact, i.e., C n and C is closed then there exists a 1 1 2K 2 strictly separating hyperplane H(a,b),i.e.,C1 H (a,b) and C2 ⇢ ✓ H (a,b) and C1 H(a,b)= = C2 H(a,b). ≥ \ ; \ 1. Basic (and some more) facts from Convexity 3 H (a,b) H (a,b) ≥ C2 C1 H(a,b) Figure 1.2: Strictly separating hyperplane of two convex bodies A hyperplane H(a,b) passing through the boundary of a closed convex set C and containing C entirely in one of the two halfspaces, i.e., C \ H(a,b) = and C H (a,b), say, is called a supporting hyperplane of C. 6 ; ✓ The intersection of F = C H(a,b) is called a face of C, or more precisely, \ a k-face if dim F = k.Ifk =0thenF is called an extreme point of C. Faces themselves are closed convex sets, and the interesection of faces of C is again a face of C, where we regard the empty set as the ( 1)-dimensional − face of C. The boundary of C, denoted by bd C, is the union of its faces. For a convex body K n and a given normal vector (direction) a n 2K 2 R 0 the supporting hyperplane in direction a is given by H(a,hK (a)) \{ } where n hK : R R with hK (a) := max a, x : x K ! {h i 2 } is called the support function of K. The support function is positive homoge- neous,i.e.,hK (λ a)=λhK (a), λ R 0, and sub-additive,i.e.,hK (x + y) n 2 ≥ hK (x)+hK (y) for all x, y R . Indeed, these two properties characterize 2 support functions of convex bodies among the real-valued functions on Rn. A quite general concept in di↵erent branches of mathematics is that of duality and/or polarity. In our setting it is defined as follows: for X Rn, ✓ the set ? n X := y R : x, y 1 for all x X 2 h i 2 is called the polar set of X. As the intersection of convex closed sets con- taining the origin, the polar set is always a convex closed set containing 0. Its actual shape, however, depends on its position relative to the origin. For instance, X? is bounded if and only if 0 is an interior point of X. On the other hand, as stated in the next proposition, its shape be- haves properly with respect to linear transformations and a polar set of a o-symmetric convex body is again an o-symmetric convex body. 1.2 Proposition. n ? ? i) Let X R , and let M GL(n, R).Then(MX) = M −|X . Here ✓ 2 GL(n, R) denotes the general linear group of all invertible real n n ⇥ 4 1. Basic (and some more) facts from Convexity 1/2 ? ? 1/2 0 0 0 0 1/2 1/2 − 1 1 1 Figure 1.3: Polar bodies of B2 and 0 + B2 matrices. ? ii) Let K Rn be convex and closed with 0 K.Then(K?) = K. ✓ 2 n ? n iii) Let K o .ThenK o and x K? = maxy K x, y = hK (x). 2K 2K | | 2 h i By Proposition 1.2 iii) and H¨older’s inequality for sums, n n xk yk x p y p/(p 1) , x, y R ,p (1, ), | || | | | − 2 2 1 Xi=1 with equality if and only if x =c y p 1 for a constant c, the polar bodies | i| | i| − of the p-norms are given by p p ? p 1 1 ? ? 1 (B ) = Bn− , 1 <p< , and (B ) = B1 and thus (B1) = B . n 1 n n n n Since we are mainly dealing with simple structured bounded convex sets, we define the volume of sets via the Jordan-Peano measure: A subset D n n ⇢ R is called (Jordan-Peano) measurable if its indicator function χD : R ! 0, 1 given by { } 1, x D, χD(x):= 2 (0, x / D, 2 is Riemann integrable, i.e., the Riemann integral vol(D):= χD(x)dx n ZR exists. vol(D) is called the volume of D. So the volume of a set D is based on the principle of exhausting or approximating D by ”small” n-dimensional n boxes x R : ↵i xi βi, 1 i n , ↵i <βi R, for which the { 2 } 2 volume is defined as n (β ↵ ). In particular, we have V(X) = 0 for a i=1 i − i measureable set of dim X<n, and Q 1.3 Proposition. Let X Rn be a measurable set. Then ⇢ #(X 1 Zn) vol(X)= lim \ m , m n !1 m 1. Basic (and some more) facts from Convexity 5 n n where Z = z R : zi Z denotes the set of all points with integral { 2 2 } coordinates. 1 n Figure 1.4: Approximation by ”small” boxes centered at points of m Z The unit ball Bn1, i.e., the cube (box) of edge length 2 centered at the n p origin has vol(Bn1)=2 .
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