A note on the honeycomb optimality among periodic convex tilings

TL;DR

This paper proves that the hexagonal honeycomb is the optimal convex periodic tiling of the plane under certain cost conditions, extending the understanding of optimal tessellations.

Contribution

It establishes the honeycomb's optimality among convex periodic tilings under specific semicontinuity and symmetrization conditions.

Findings

Hexagonal honeycomb is optimal among convex periodic tessellations.

Optimality holds under lower semicontinuity and Steiner symmetrization conditions.

Provides theoretical justification for the honeycomb's efficiency in tiling.

Abstract

We show that the hexagonal honeycomb is optimal among convex periodic tessellations of the plane, provided the cost functional is lower semicontinuous with respect to the Hausdorff convergence, and decreasing under Steiner symmetrization.