On antipodal and adjoint pairs of points for two convex bodies (Q1892437)

Let \(K_1\), \(K_2\) be two disjoint convex bodies in Euclidean \(d\)-space. A pair of points \(x_1 \in K_1\), \(x_2 \in K_2\) is called strictly adjoint if there exist parallel hyperplanes \(H_1\) through \(x_1\) and \(H_2\) through \(x_2\) both separating \(K_1\) and \(K_2\) such that \(H_i \cap K_i = \{x_i\}\) for \(i = 1,2\). Similarly, \(x_1 \in K_1\) and \(x_2 \in K_2\) are called strictly antipodal if for \(i = 1,2\) there exist hyperplanes \(H_i\) through \(x_i\) such that \(K_1\) and \(K_2\) both lie in the same half-space determined by \(H_i\) and \(H_i \cap K_i = \{x_i\}\). Let \(\overline{p}(K_1,K_2)\) \([\overline{q}(K_1,K_2)]\) denote the number of strictly antipodal [strictly adjoint] pairs of points of \(K_1\) and \(K_2\). The author shows that for any two \(K_1\), \(K_2\) there exists at least one pair of adjoint points, i.e. \(\overline{p} (K_1, K_2) \geq 1\), and at least \(d\) pairs of antipodal points, i.e. \(\overline{q} (K_1, K_2) \geq d\). He further shows: If \(\overline {p}(K_1, K_2')\) is finite for every translate \(K_2'\) of \(K_2\) then \(K_1\) and \(K_2\) are polytopes, and vice versa. The same result is true if \(\overline{p}\) is replaced by \(\overline{q}\). There are some more results in the same spirit.