A Discrete KKT Variational Characterization of the Local Minimality of the Mahler Volume in Centrally Symmetric Polytopes

Abstract

We present a discrete parametric characterization of the Mahler functional $\mathcal{V}_M$ for centrally symmetric polytopes in $\mathbb{R}^n$. By formulating the first variation of the volume with respect to the radial immersions of the vertices, we derive an exact KKT stationarity condition. Through this formalism, we demonstrate that the Hanner orbit constitutes a strict local minimum in the stratum of radial dilations modulo $GL(n, \mathbb{R})$, for all dimensions $n \ge 2$. Spectral analysis of the second variation reveals that the radial Hessian matrix is analytically equivalent to a positive semi-definite discrete Graph Laplacian. Coupling this isochoric result with a simplicial truncation analysis in the Hausdorff topology, we establish a quadratic quantitative stability bound against local polyhedral perturbations. This discrete framework eludes the degenerations of traditional smooth continuous analysis, providing an explicit algebraic resolution for the strict local minimality of this geometric configuration and its topological isolation in the subspace of polytopes.