Hypersurfaces with central convex cross-sections

TL;DR

This paper characterizes hypersurfaces in higher-dimensional space that have the property that their cross-sections with certain hyperplanes are centrally symmetric ovaloids, showing they are either cylinders over such ovaloids or quadrics.

Contribution

It provides a classification of hypersurfaces with the central ovaloid property in dimensions four and higher, identifying them as either cylinders or quadrics.

Findings

Hypersurfaces with the central ovaloid property are either cylinders over a central ovaloid or quadrics.

The property imposes strong geometric constraints leading to this classification.

Complete, connected, smooth hypersurfaces with this property must be one of these two types.

Abstract

A hypersurface $M$ in $\mathbb{R}^n$, $n \geq 4$, has central ovaloid property if $M$ intersects some hyperplane transversally along an ovaloid and every such ovaloid on $M$ has central symmetry. We show that a complete, connected, smooth hypersurface with central ovaloid property must either be a cylinder over a central ovaloid or else quadric.