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On the edge densities of normal, convex mosaics

Máté Kadlicskó, Zsolt Lángi, Shanxiang Lyu

TL;DR

The paper addresses the minimal edge density in normal, convex mosaics by first solving the decomposable 2D-structure case, deriving exact lower bounds with explicit equality configurations across dimensions. It then focuses on 3D translative mosaics, showing that cubes minimize edge density among unit-volume translative mosaics and connecting these results to a broader optimization over 3D parallelohedra via belt structures. A central engine is a two-tier approach: (i) reduce to surface-isotropic position and (ii) bound a belt-weighted objective $w_m(P)$ using Cauchy–Schwarz, yielding a complete classification of minimizers among the five parallelohedron types. The findings advance understanding of Kelvin-Bezdek-type edge-density problems in higher dimensions and provide a framework for translating edge-length optimization into fixed-volume mosaic geometry.

Abstract

In this paper we investigate the problem of finding the minimum edge density in families of convex, normal mosaics with unit volume cells in $n$-dimensional Euclidean space. In the first part of the paper we solve this problem for mosaics whose cells are Minkowski sums of cells of $1$ or $2$-dimensional mosaics. We show that while for $n=2$ this minimum is attained by a mosaic with regular hexagon cells, this is not true in any dimension $n > 2$, where the minimum is attained by a mosaic whose cells are Minkowski sums of pairwise orthogonal regular triangles, and possibly a segment. In the second part we investigate $3$-dimensional convex mosaics whose cells are translates of a given convex polyhedron, and show that within this family, mosaics with cubes as cells have minimum edge density. In addition, using our method, in the family of $3$-dimensional convex polyhedra whose translates tile the space, we find the unit volume polyhedra with minimal total edge length.

On the edge densities of normal, convex mosaics

TL;DR

The paper addresses the minimal edge density in normal, convex mosaics by first solving the decomposable 2D-structure case, deriving exact lower bounds with explicit equality configurations across dimensions. It then focuses on 3D translative mosaics, showing that cubes minimize edge density among unit-volume translative mosaics and connecting these results to a broader optimization over 3D parallelohedra via belt structures. A central engine is a two-tier approach: (i) reduce to surface-isotropic position and (ii) bound a belt-weighted objective using Cauchy–Schwarz, yielding a complete classification of minimizers among the five parallelohedron types. The findings advance understanding of Kelvin-Bezdek-type edge-density problems in higher dimensions and provide a framework for translating edge-length optimization into fixed-volume mosaic geometry.

Abstract

In this paper we investigate the problem of finding the minimum edge density in families of convex, normal mosaics with unit volume cells in -dimensional Euclidean space. In the first part of the paper we solve this problem for mosaics whose cells are Minkowski sums of cells of or -dimensional mosaics. We show that while for this minimum is attained by a mosaic with regular hexagon cells, this is not true in any dimension , where the minimum is attained by a mosaic whose cells are Minkowski sums of pairwise orthogonal regular triangles, and possibly a segment. In the second part we investigate -dimensional convex mosaics whose cells are translates of a given convex polyhedron, and show that within this family, mosaics with cubes as cells have minimum edge density. In addition, using our method, in the family of -dimensional convex polyhedra whose translates tile the space, we find the unit volume polyhedra with minimal total edge length.
Paper Structure (5 sections, 7 theorems, 49 equations, 2 figures, 1 table)

This paper contains 5 sections, 7 theorems, 49 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:decomp']}
  3. Preliminaries for the proofs of Theorems \ref{['thm:1']} and \ref{['thm:totaledgelength']}
  4. The proof of Theorems \ref{['thm:1']} and \ref{['thm:totaledgelength']} using Theorem \ref{['thm:2']}
  5. The proof of Theorem \ref{['thm:2']}

Key Result

Theorem 1

Let $n \geq 2$, and let $\mathcal{M} \in \mathcal{F}_{dc}^n$. Then Furthermore, in the above inequality we have equality

Figures (2)

  • Figure 1: Combinatorial types of $3$-dimensional parallelohedra. The type (5) parallelohedron is the regular truncated octahedron generated by the six segments connecting the midpoints of opposite edges of a cube. The type (3) polyhedron is the regular rhombic dodecahedron generated by the four diagonals of the same cube. The other parallelohedra in the picture are obtained by removing some generating segments from the type (5) parallelohedron.
  • Figure 2: The lower bounds for $w_m^i$ from the second column of Table \ref{['tab:wmP']} as functions of $\alpha_4$, with $\alpha_6 = 1$, and $0 < \alpha_4 \leq 1$.

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Conjecture 1
  • Theorem 4
  • Lemma 1
  • Lemma 2
  • proof
  • ...and 2 more