On the Space of Convex Figures

Abstract

Let T be the set of convex bodies in ℝ^k, and let T be the set of their similarity classes. If k = 2, then F is written instead of T. Ametric d on T is defined by setting d({K₁}, {K₂}) = inf{log(b/a)} for the classes {K₁}, {K₂} ∈ T of convex bodies K₁ and K₂, where a and b are positive reals such that there is a similarity transformation A with aA(K₁) ⊂ K₂ ⊂ bA(K₁). Let D₂ be the planar unit disk. If x > 0, then F_x denotes the set of planar convex figures K in F with d({D₂}, {K}) ≥ x. In addition, the sets T and F are equipped with the usual Hausdorff metric. It is proved that if y > log(sec(π/n)) ≥ x for a certain integer n greater than 2, then no mapping F_x → F_y is SO(2)-equivariant. Let M_k denote the space of k-dimensional convex polyhedra and let M_k(n) ⊂ M_k be the space of polyhedra with at most n hyperfaces (vertices). It is proved that there are no SO(k)-equivariant continuous mappings M_k(n + k) → M_k(n). Let T ^s be the closed subspace of T formed by centrally symmetric bodies. Let T_x denote the closed subspace of T formed by the bodies K with d(T ^s, {K}) ≥ x > 0. It is proved that for each positive y there exists a positive x such that no mapping T_x → T_y is SO(k)-equivariant. Bibliography: 3 titles.