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. Spectral analysis of the second variation shows that the radial Hessian matrix is analytically equivalent to a positive semi-definite discrete Graph Laplacian. Coupling this radial analysis with a combinatorial study of isovertex folding and vertex truncations in the local Hausdorff topology, we establish a quadratic quantitative stability bound against general polyhedral perturbations. This discrete framework avoids the degenerations of traditional continuous analysis and provides an explicit algebraic proof that the Hanner orbit constitutes a strict and topologically isolated local minimum in the space of centrally symmetric polytopes modulo $GL(n,\mathbb{R})$.