顶点插入可增加多面体的金字塔宽度，打破长期猜想。
Pyramidal Width Can Increase Under Vertex Insertion
- 通过构造反例验证顶点添加后金字塔宽度可能增大。
- 实测宽度平方从48/353增至36/133，提升约41.1%。
- 适用于凸几何与优化理论研究者，验证方法可信赖。
Lacoste-Julien与Jaggi在2015年提出猜想：若多面体所有旧点仍为顶点，则添加新顶点不会增加其金字塔宽度。本文给出一个精确反例，包含ℝ³中六个整数点。设P=conv{v₀,…,v₄}，Q=conv{v₀,…,v₅}，其中各顶点坐标为：v₀=(-1,-3,-1)，v₁=(3,2,-2)，v₂=(0,2,1)，v₃=(-1,-3,3)，v₄=(-2,0,1)，v₅=(-1,0,-2)。P的五个顶点在Q中仍为顶点，但P的金字塔宽度平方为48/353，而Q为36/133。因此，金字塔宽度提升因子约为√(1059/532)≈1.410886779。证明基于金字塔宽度与面距离的等价性，通过整数支撑超平面验证两个多面体的面格结构，并以有限有理计算评估所有面距离。论文附带独立的精确验证工具。
原文摘要 · Abstract (English)
Lacoste-Julien and Jaggi conjectured in 2015 that the pyramidal width of a polytope cannot increase when a vertex is added, provided that every old point remains a vertex. We give an exact counterexample with six integer points in $\R^3$. For \[ P=\conv\{v_0,\ldots,v_4\},\qquad Q=\conv\{v_0,\ldots,v_5\}, \] where \[ \begin{aligned} v_0&=(-1,-3,-1), & v_1&=(3,2,-2), & v_2&=(0,2,1),\\ v_3&=(-1,-3,3), & v_4&=(-2,0,1), & v_5&=(-1,0,-2), \end{aligned} \] all five vertices of $P$ remain vertices of $Q$, but \[ \PWidth(P)^2=\frac{48}{353} \quad\text{and}\quad \PWidth(Q)^2=\frac{36}{133}. \] Thus vertex insertion increases pyramidal width by the factor $\sqrt{1059/532}\approx 1.410886779$. The proof uses the equivalence between pyramidal width and facial distance, certifies both face lattices by integer supporting hyperplanes, and evaluates every facial distance by a finite rational calculation. A dependency-free exact verifier accompanies the paper.
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