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arXiv · 2609.12640

Non-convex unit-edge polytopes on kissing configurations in dimensions 5-7

Matthew Self
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Abstract

All nine known conjecturally optimal non-lattice kissing configurations in dimensions 5, 6, and 7 are the vertex sets of polytopes with only unit edges. Eight of the nine are non-convex, and the contact polytopes, the convex hulls of the same points, have longer edges. The edges of the unit-edge polytopes are exactly the contacts of the configuration. All but two of the nine are constructed from the lattice contact polytope in the same dimension by splitting some of its facets and reassembling the pieces. This is a geometric construction, distinct from the constructions by layers. The facets that fold when split are consecutive members of the Gosset series $k_{21}$, and a split in dimension $n$ folds to the inner product $1/(10-n)$: $1/5$, $1/4$, or $1/3$. These are the inner products by which the contact polytopes differ from the lattice one. All 12 unit-edge polytopes, the nine and the three lattice ones, are creased, a class we define that extends convexity by admitting shallow folds that reach no more than halfway. Each is the only creased unit-edge polytope on its vertex set. Every statement is certified in exact arithmetic.

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BibTeXRIS

Matthew Self. 2026-09-11. Non-convex unit-edge polytopes on kissing configurations in dimensions 5-7. https://arxiv.org/abs/2609.12640

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