Sections and projections of nested convex bodies

TL;DR

This paper explores new characterizations of Euclidean balls and ellipsoids using properties of nested convex bodies, their sections, projections, and support cones, extending classical results in Geometric Tomography.

Contribution

It introduces novel characterizations of ellipsoids based on sections and projections of nested convex bodies, a setting not previously studied.

Findings

Characterizations of ellipsoids via nested convex bodies

New relations between sections, projections, and support cones

Results applicable to boundary apex support cones

Abstract

One of the most important problems in Geometric Tomography is to establish properties of a given convex body if we know some properties over its sections or its projections. There are many interesting and deep results that provide characterizations of the sphere and the ellipsoid in terms of the properties of its sections or projections. Another kind of characterizations of the ellipsoid is when we consider properties of the support cones. However, in almost all the known characterizations, we have only a convex body and the sections, projections, or support cones, are considered for this given body. In this article we proved some results that characterizes the Euclidean ball or the ellipsoid when the sections or projections are taken for a pair of nested convex bodies, i.e., two convex bodies $K$, $L$ such that $L\subset\text{int}\, K.$ We impose some relations between the corresponding sections or projections and some apparently new characterizations of the ball or the ellipsoid appear. We also deal with properties of support cones or point source shadow boundaries when the apexes are taken in the boundary of $K$.