A Spectral Approach to Polytope Diameter
summary
Constructs a weighted 1-skeleton from the Hessian of log-volume of the polar polytope, shows that Hessian has exactly one positive eigenvalue and decomposes as minus a Laplacian plus a positive diagonal, and proves a spectral gap lower bound roughly spectral gap ~ sin^2(min angle between facets). Using Chebyshev polynomial amplification on the generalized adjacency yields a diameter bound that replaces prior d^3 dependence by a milder factor (roughly log m times delta in the stated regime), giving a concrete spectral-to-diameter pipeline.
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