Pyramidal Width Can Increase Under Vertex Insertion

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 ${\mathbb{R}}^3$. For $$P=\text{conv}\{v_0,\dots ,v_4\},Q=\text{conv}\{v_0,\dots ,v_5\},$$ where $$\begin{array}{rlrlrl}{\displaystyle v_0}& {\displaystyle =(-1,-3,-1),}& {\displaystyle v_1}& {\displaystyle =(3,2,-2),}& {\displaystyle v_2}& {\displaystyle =(0,2,1),}\\ {\displaystyle v_3}& {\displaystyle =(-1,-3,3),}& {\displaystyle v_4}& {\displaystyle =(-2,0,1),}& {\displaystyle v_5}& {\displaystyle =(-1,0,-2),}\end{array}$$ all five vertices of $P$ remain vertices of $Q$, but $$\text{PWidth}(P{)}^2=\frac{48}{353}\text{and}\text{PWidth}(Q{)}^2=\frac{36}{133}\mathrm{.}$$ Thus vertex insertion increases pyramidal width by the factor $\sqrt{1059\mathrm{/}532}\approx 1.410886779$. 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.