Drums of high width

TL;DR

The paper constructs a new infinite family of 5-dimensional drums $D_k$ whose width in the facet-ridge graph grows linearly with the number of vertices ($n=16k+24$). By exploiting a rich symmetry group and a detailed analysis of the interaction between the top and bottom drum skins through the facet-vertex map and the pair embedding, it proves $\text{width}(D_k) \ge 5+k$, translating (via Santos’ strong d-step theorem) into high-dimensional polytopes with diameter exceeding the Hirsch bound by $k$. Consequently, the Hirsch excess for these constructions tends to $1/16$, the largest known value, and provides an explicit mechanism for achieving linear excess in dimension. The work also outlines a practical, symmetry-driven approach to constructing and verifying such drums, connecting combinatorial geometry with linear-programming-inspired methods to bound widths. This advances understanding of polytope diameters and offers concrete counterexamples with nearly optimal excess width growth.

Abstract

We provide a family of $5$-dimensional prismatoids whose width grows linearly in the number of vertices. This provides a new infinite family of counter-examples to the Hirsch conjecture whose excess width grows linearly in the number of vertices, and answers a question of Matschke, Santos and Weibel.

Figures (5)

• Figure 1: Our points $m_{\pm\pm}$, $p_{\pm}$ and $a_i$ for $k = 5$. (Not to scale!)
• Figure 2: Projection of the facet $B$.
• Figure 3: Projection of the facet $E_i$.
• Figure 4: Projection of the facet $C_i$.
• Figure 5: The subgraph ${\mathcal{G}}$.

Theorems & Definitions (49)

• Theorem 1.1
• Corollary 1.2
• Remark 1.3
• Remark 1.4
• Remark 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• Proposition 2.4
• proof