Many equiprojective polytopes

TL;DR

The paper advances the study of $3$-dimensional equiprojective polytopes by establishing a near-quadratic growth lower bound on the number of combinatorial types, namely at least $k^{rac{3}{2}k+o(k)}$ as $k oty$. It achieves this through two complementary constructions: (i) a zonotope-based approach that yields the even-$k$ case via the known fact that a $3$-D zonotope with $n$ generators is $2n$-equiprojective and (ii) a Minkowski-sum framework using aggregated cones to handle odd $k$ by adjoining a triangle to a zonotope, producing a $(2n+3)$-equiprojective polytope. The aggregated-cone machinery provides a robust criterion for equiprojectivity and a practical method to analyze how Minkowski sums affect $k$, enabling new classes of equiprojective polytopes and guiding decomposability questions. The results illuminate Shephard's question by showing rich, constructive families of equiprojective polytopes and highlight open problems about indecomposability within this class.

Abstract

A $3$-dimensional polytope $P$ is $k$-equiprojective when the projection of $P$ along any line that is not parallel to a facet of $P$ is a polygon with $k$ vertices. In 1968, Geoffrey Shephard asked for a description of all equiprojective polytopes. It has been shown recently that the number of combinatorial types of $k$-equiprojective polytopes is at least linear as a function of $k$. Here, it is shown that there are at least $k^{3k/2+o(k)}$ such combinatorial types as $k$ goes to infinity. This relies on the Goodman--Pollack lower bound on the number of order types and on new constructions of equiprojective polytopes via Minkowski sums.

Figures (4)

• Figure 1: A triangular prism (left), its orthogonal projection along $u$ (center), and its aggregated cone $C_P(u)$ at $u$ (right) for two edge directions $u$ (top and bottom).
• Figure 2: A regular dodecahedron $P$ (left), its orthogonal projection along the edge direction $u$ parallel to the edges $e$ and $e'$ (center), and its aggregated cone $C_P(u)$ at $u$ (right).
• Figure 4: A triangle in $\mathbb{R}^3$ (colored gray) and its normal cones at edges (colored green) and at itself (black vertical line).
• Figure 5: The equitruncated tetrahedron from HasanHossainLopez-OrtizNusratQuaderRahman2022.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1
• Proposition 2.2
• Theorem 2.3
• Definition 3.1
• Theorem 3.2
• Definition 3.3