Characterizations of generalized convex bodies of revolution

TL;DR

This paper establishes that certain symmetry sequences in convex bodies of revolution in higher-dimensional Euclidean spaces are sufficient to characterize them as generalized bodies of revolution, sometimes even spheres.

Contribution

It proves that sequences of axes or hyperplanes of symmetry can determine a convex body as a generalized body of revolution, extending previous symmetry characterizations.

Findings

Symmetry sequences imply the body is a generalized body of revolution.

In some cases, symmetry sequences imply the body is a sphere.

Provides new symmetry-based characterizations in convex geometry.

Abstract

In this work we prove that either a sequence of axes of symmetry or a sequence of hyperplanes of symmetry of a convex body $K$ in the Euclidean space $E^d, d>2$, are enough to guarantee that $K$ is a generalized body of revolution (and in some cases a sphere).