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arXiv · 2608.06897

A Combinatorial Approach to Vorono{\"i} Theory

Romain M{é}nab{é}
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Abstract

This article introduces crystallographic wraiths, new combinatorial invariants attached to Euclidean lattices and linked to the geometry of Vorono{\"i} cells. They are based on geometric properties of shortest linear dependencies among relevant Vorono{\"i} vectors. Moreover, it explores the new theory of crystallographic wraiths, proves that only finitely many occur in each dimension, and extends the construction to a larger class of combinatorial objects. As an application, it provides a new combinatorial proof of Vorono{\"i}'s finiteness theorem for perfect lattices.

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Romain M{é}nab{é}. 2026-08-07. A Combinatorial Approach to Vorono{\"i} Theory. https://arxiv.org/abs/2608.06897

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