发表机构
University of California, San Diego（加利福尼亚大学圣迭戈分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对2015年的棱锥宽度不会因顶点插入而增大的猜想，构造三维整数多面体反例，证明顶点插入可增大棱锥宽度，相关证明附带精确验证工具。
AI 中文摘要
Lacoste-Julien与Jaggi于2015年提出猜想：当添加顶点且所有旧点仍为顶点时，多面体的棱锥宽度不会增大。我们在三维欧氏空间R³中给出含6个整数点的精确反例：令P为v₀至v₄的凸包，Q为v₀至v₅的凸包，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5个顶点均保留为Q的顶点，但P的棱锥宽度平方(PWidth(P)²)为48/353，Q的棱锥宽度平方(PWidth(Q)²)为36/133，顶点插入使棱锥宽度增大了√(1059/532)≈1.410886779倍。证明利用棱锥宽度与面距离的等价性，通过整数支撑超平面验证面格，以有限有理计算得到各面距离，论文附带无依赖的精确验证工具。
英文摘要
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.
