On the three-dimensional Blaschke-Lebesgue problem

TL;DR

This paper investigates the Blaschke-Lebesgue problem in three dimensions, providing a necessary condition for the shape of volume-minimizing convex sets of fixed width, revealing they must have boundary components with constant smaller principal curvature.

Contribution

It establishes a necessary geometric condition for minimizers in the three-dimensional Blaschke-Lebesgue problem, linking boundary curvature properties to shape characteristics.

Findings

Boundary components have constant smaller principal curvature

Minimizers' boundary are spherical caps or canal surfaces

Provides geometric constraints for shape optimization in fixed width sets

Abstract

The width of a closed convex subset of Euclidean space is the distance between two parallel supporting planes. The Blaschke-Lebesgue problem consists of minimizing the volume in the class of convex sets of fixed constant width and is still open in dimension n > 2. In this paper we describe a necessary condition that the minimizer of the Blaschke-Lebesgue must satisfy in dimension n=3: we prove that the smooth components of the boundary of the minimizer have their smaller principal curvature constant, and therefore are either spherical caps or pieces of tubes (canal surfaces).