Unique local determination of convex bodies

Makai, Endre and Martini, H. (2016) Unique local determination of convex bodies. Acta Mathematica Hungarica, 150 (1). pp. 176-193. ISSN 0236-5294 (print), 1588-2632 (online)

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Abstract

Barker and Larman asked the following. Let K′⊂ Rd be a convex body, whose interior contains a given convex body K⊂ Rd, and let, for all supporting hyperplanes H of K, the (d − 1)-volumes of the intersections K′∩ H be given. Is K′ then uniquely determined? Yaskin and Zhang asked the analogous Question when, for all supporting hyperplanes H of K, the d-volumes of the “caps” cut off from K′ by H are given. We give local positive answers to both of these questions, for small C2-perturbations of K, provided the boundary of K is C+ 2. In both cases, (d − 1)-volumes or d-volumes can be replaced by k-dimensional quermassintegrals for 1 ≤ k≤ d- 1 or for 1 ≤ k≤ d, respectively. Moreover, in the first case we can admit, rather than hyperplane sections, sections by l-dimensional affine planes, where 1 ≤ k≤ l≤ d- 1. In fact, here not all l-dimensional affine subspaces are needed, but only a small subset of them (actually, a (d − 1)-manifold), for unique local determination of K′. © 2016, Akadémiai Kiadó, Budapest, Hungary.

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