Pessimal packing shapes

Kallus, Yoav

doi:10.2140/gt.2015.19.343

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Pessimal packing shapes

Yoav Kallus

Geometry & Topology 19 (2015) 343–363

DOI: 10.2140/gt.2015.19.343

Abstract

We address the question of which convex shapes, when packed as densely as possible under certain restrictions, fill the least space and leave the most empty space. In each different dimension and under each different set of restrictions, this question is expected to have a different answer or perhaps no answer at all. As the problem of identifying global minima in most cases appears to be beyond current reach, in this paper we focus on local minima. We review some known results and prove these new results: in two dimensions, the regular heptagon is a local minimum of the double-lattice packing density, and in three dimensions, the directional derivative (in the sense of Minkowski addition) of the double-lattice packing density at the point in the space of shapes corresponding to the ball is in every direction positive.

Keywords

packing, convex body, lattice, density

Mathematical Subject Classification 2010

Primary: 52A40

Secondary: 52C15, 52C17

References

Publication

Received: 11 August 2013

Accepted: 5 June 2014

Published: 27 February 2015

Proposed: Dmitri Burago

Seconded: Walter Neumann, Yasha Eliashberg

Authors

Yoav Kallus
Santa Fe Institute 1399 Hyde Park Road Santa Fe, NM 87501 USA

Volume 19, issue 1 (2015)