Characterizations of generalized convex bodies of revolution

TL;DR

The paper addresses when partial symmetry patterns force a convex body $K\subset \mathbb{E}^{d}$, $d\ge 3$, to be a generalized body of revolution or a sphere, by introducing and analyzing $k$-axes of symmetry, $k$-planes of symmetry, and sequences of symmetries. It proves two core results: (i) if a $k$-flat $\Lambda$ through $p$ has every line through $p$ in $\Lambda$ as an axis, then $K$ is a $k$-body of revolution, and (ii) if a $(k+1)$-flat $\Lambda$ through $p$ makes every $k$-flat through $p$ in $\Lambda$ a $k$-axis, then $K$ is a $(k+1)$-body of revolution; it also shows that a sequence of axes or a sequence of hyperplanes of symmetry suffices to force $K$ to be a $k$-body of revolution or a sphere. The authors develop a unified framework using $n$-stars of lines, limit/stability arguments for symmetry, circumsphere-based centrality, and classical tools like the Süss–Schneider theorem to derive a suite of results that extend known 3D phenomena to higher dimensions. These findings contribute to the broader understanding of how partial symmetry controls the global geometric structure of convex bodies, with potential implications for convex geometry and symmetry classification.

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).

Theorems & Definitions (23)

• Theorem 1
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• Theorem 3
• Theorem 4
• Theorem 5
• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5