Sections and projections of nested convex bodies

TL;DR

The paper addresses ellipsoid and ball recognition from sectional and projection data in the setting of nested convex bodies $L \subset \mathrm{int}\,K$. It fuses floating-body theory, $n$-cycles of hyperplanes, and support-cone geometry to obtain rigidity results: under specific section- or projection-relations between $K$ and $L$, the pair must be ellipsoids (or $K$ must be a ball in odd dimensions), and it extends classical Blaschke–Olovjanishnikov–Gruber–Ódor characterizations to nested configurations and grazing data. It also provides exact 2D and odd-dimensional results, plus multiple conjectures and open problems that connect section/projection data to ellipsoidal rigidity. Collectively, these results deepen geometric tomography by showing how nested cross-sections and shadow data constrain convex bodies and enable more robust shape reconstruction.

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 $K$.

Figures (6)

• Figure 1: The floating body
• Figure 2: The point $x$ projects onto the midpoint of $[a',b']$
• Figure 4:
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Theorems & Definitions (13)

• Conjecture 1
• Lemma 1
• Theorem 1
• Conjecture 2
• Conjecture 3
• Remark 1
• Remark 2
• Theorem 2
• Theorem 3
• Theorem 4