Volumes of consecutively defined sets

Abstract

We study a variant of the graph polytopes of a path and of a cycle where we replace the inequality $x_i+x_{i+1}\le 1$ with the two inequalities $(1-\alpha )\cdot x_i+\alpha \cdot x_{i+1}\le \alpha$ for $0\le x_i\le \alpha$ and $\alpha \cdot x_i+(1-\alpha )\cdot x_{i+1}\le \alpha$ for $\alpha \le x_i\le 1$. Using a self-adjoint operator and its eigenvalues we obtain convergent series for their volumes. As a corollary we obtain that the volumes of the set associated to a path on $n$ vertices and the set associated to a cycle on $n+1$ vertices are related by a constant factor of $\alpha$.

9 pages, 1 figure