Infinite curvature on typical convex surfaces

TL;DR

This paper proves that most convex surfaces have points with infinite curvature in every tangent direction, using a new curvature definition and approximation techniques.

Contribution

It introduces a novel curvature concept inspired by Alexandrov spaces and demonstrates its continuity, solving a long-standing open problem in convex geometry.

Findings

Typical convex surfaces contain points of infinite curvature in all tangent directions.

A new curvature definition with favorable continuity properties is established.

Convex surfaces can be approximated by smooth surfaces while preserving curvature properties.

Abstract

Solving a long-standing open question in convex geometry, we will show that typical convex surfaces contain points of infinite curvature in all tangent directions. To prove this, we use an easy curvature definition imitating the idea of Alexandrov spaces of bounded curvature, and show continuity properties for this notion. Along the way, we show a theorem for the approximation of convex surfaces by smooth surfaces.