On antipodal and adjoint pairs of points for two convex bodies

Discrete Comput Geom Geometry Discrete & Computational On Antipodal 0 Adjoint Pairs of Points for Two Convex Bodies 0 V. Soltan 0 0 Mathematical Institute of the Academy of Sciences of Moldova , Str. Academiei nr. 5, Chi~in~u277028, Republica Moldova The numbers of antipodal and of adjoint pairs of points are estimated for a given pair of disjoint convex bodies in E d. 1. I n t r o d u c t i o n It is well known (see, for instance, [4]) that any two disjoint convex bodies K t , K 2 in the Euclidean space E a can be strictly separated by a hyperplane H, i.e., K j , K 2 lie in distinct open half-spaces determined by H. This result easily implies the existence of two distinct parallel hyperplanes H I , H 2 both separating K1, K 2 such that H~ supports K~ and H 2 supports K 2. The last assertion has b e e n improved by De Wilde [8], who showed that the above hyperplanes H~, H 2 can be chosen so that the sets of contact H~ A K~, H 2 n K 2 are single points. Based on this result, we introduce the following definition. (As usual, exp K and ext K denote, respectively, the set of exposed points and the set of extreme points of K.) D e f i n i t i o n 1. Let K1, K 2 be disjoint convex bodies in E d. We say that points x 1 E ext K 1 and x 2 E ext K 2 are adjoint if there are distinct parallel hyperplanes H1, H 2 through Xl, x2, respectively, both separating K 1 and K 2. If, additionally, H 1 A K 1 = {x1} and H 2 N K 2 = {x2}, the points x t , x 2 are called strictly adjoint. - D e f i n i t i o n 2. Let K1, K 2 be disjoint convex bodies in E d. We say that points x I ~ ext K 1 and x 2 ~ ext K 2 are antipodal provided there are parallel hyperplanes H1, H 2 through x 1, x2, respectively, such that both K1, K 2 lie between H1, H E. If, additionally, H 1 A K 1 = { x 1} and H 2 A K 2 = { x 2 } , the points x l , x 2 are called strictly antipodal. Clearly, extreme points xa E Ka, x 2 ~ K 2 forming a strictly antipodal or strictly adjoint pair are exposed for K~, K 2, respectively. In our notation D e Wilde's theorem states that any two disjoint convex bodies in E ~ determine at least one strictly adjoint pair of points. O u r purpose here is to sharpen D e Wilde's result and to prove a few related assertions on the numbers of (strictly) adjoint and of (strictly) antipodal pairs determined by two disjoint translates of a given pair of convex bodies. For similar results on the numbers of antipodal pairs and strictly antipodal pairs of points o f a single convex body in E a see [6]. Main Results Denote by p ( K l , K 2) (by p ( K 1 , K2)) the number of antipodal (strictly antipodal) pairs of points x 1 ~ K1, x z ~ K s. Similarly, denote by q ( K 1 , K 2) (by gI(Ka, K2)) the number o f adjoint (strictly adjoint) pairs of points x 1 E Ka, x 2 ~ K 2. Here and subsequently, we mean that two pairs {xl, x2}, {X'l, x'2} of points, where x l , x] ~ K 1 and x2, x~ ~ K2, are distinct if either x I 4=x] or x 2 4=x~. Define any of the values p ( K 1, K2), ~ ( K 1, K2), q ( K 1, K2), ~/(K1, K 2) to be ~ if the respective family of pairs is infinite. Clearly, p ( K l , K 2) >_/3(KI, K2) and q(K1, K 2) > q(K1, K2). Theorem 1. /3(K1, K s) >_ 1 and gI(K1, K 2) >_ d f o r any disjoint convex bodies K1, K 2 in E d. Examples 1 and 2 below demonstrate that the inequalities in T h e o r e m 1 are sharp even for the values p ( K 1 , K 2) and q ( K l , K2). Example 1. Let K 1 be the triangle with vertices x I = (0; 0), x 2 = (0; 5), and x 3 = (5; 0), and let K 2 be the triangle with vertices Yl = (4; 4), Y2 = (3; 4), and Y3 = (4; 3) in the coordinate plane E 2. There is exactly one antipodal pair of points determined by K~, K2, namely, {xl, y~}, whence p ( K 1 , K 2) = 1. Example 2. Let K 1 be the triangle with vertices x I = (0; 0), x z = (0; 5), and x 3 = (5; 0), and let K 2 be the triangle with vertices z I = (4; 4), z 2 = (4; 9), and z 3 = (9; 4) in the coordinate plane E 2. There are exactly two adjoint pairs determined by K1, K 2 , namely, {z1, x 2} and {z1, x3}, whence q ( K l , K 2) = 2. Clearly, Examples 1 and 2 can be easily modified for the higher-dimensional case. It is easily seen that the equalities /3(K1, K 2) = 1 and U/(K1,K 2) = d are satisfied only for some special pairs {K~, K2}. The following t h e o r e m shows that any pair of convex bodies K 1, K 2 can be placed by suitable translations in order to obtain bigger values of f i ( K 1, K 2) and g/(K1, K2). Theorem 2. For any convex bodies K1, K 2 in E a, d >__2, there are translates K~, K'~ o f K 2 both disjoint to K 1 such that p ( K 1 , K '2) > d + 1 and 7/(K1, K~) > d + 1. In fact, we can restrict our attention in T h e o r e m 2 to the case when both K 1 and K 2 are polytopes. T h e o r e m 3. For convex bodies K t , K 2 in E d the following conditions are equivalent: ( 1 ) p ( K ~ , K.2') is finite for et2ery translate K'2 o f K 2 disjoint to K 1. ( 2 ) /5(K1, K~) is finite f o r every translate K~ o f K 2 disjoint to K 1 . ( 3 ) q ( K ~ , K '2) is f (...truncated)