On a strong version of the Kepler conjecture

TL;DR

This paper explores a variant of the sphere packing problem, focusing on partitioning space into convex cells with unit balls to minimize average surface area, establishing a lower bound of approximately 13.8564.

Contribution

It introduces a new problem related to sphere packing and proves a lower bound on the average surface area of convex cells containing unit balls.

Findings

Average surface area is always at least 13.8564

Provides a new perspective on sphere packing related to cell shape optimization

Establishes a quantitative bound for the problem

Abstract

We raise and investigate the following problem that one can regard as a very close relative of the densest sphere packing problem. If the Euclidean 3-space is partitioned into convex cells each containing a unit ball, how should the shapes of the cells be designed to minimize the average surface area of the cells? In particular, we prove that the average surface area in question is always at least 13.8564... .