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On the Space of Convex Figures

Research paper by V. V. Makeev, N. Yu. Netsvetaev

Indexed on: 11 Jan '16Published on: 11 Jan '16Published in: Journal of Mathematical Sciences



Abstract

Let T be the set of convex bodies in ℝk, 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 Fx 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 Fx → Fy is SO(2)-equivariant. Let Mk denote the space of k-dimensional convex polyhedra and let Mk(n) ⊂ Mk be the space of polyhedra with at most n hyperfaces (vertices). It is proved that there are no SO(k)-equivariant continuous mappings Mk(n + k) → Mk(n). Let T s be the closed subspace of T formed by centrally symmetric bodies. Let Tx 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 Tx → Ty is SO(k)-equivariant. Bibliography: 3 titles.