Convex Near-Extremizers for the Multidimensional Hermite-Hadamard Inequality

Larson proved that the boundary-mean Hermite-Hadamard inequality on bounded convex domains in R^n has sharp constant n for nonnegative subharmonic functions. We construct explicit nonnegative convex near-extremizers on collapsing simplices, proving that restricting the test functions to the convex class does not improve this universal constant. The construction uses the positive part of an affine function, with its localization scale larger than the collapsing thickness. We calculate the exact ratio and a quantitative error bound, and show that the same supremum holds for smooth, strictly positive, strictly convex test functions. This resolves the boundary-mean convex subproblem in AIM Shape optimization with surface interactions, Problem 6.1, corresponding to AIM-ANALYSIS-0032 in UnsolvedMath v1.6.0. Neither volume-power normalized sharp constant is determined here; the whole four-constant record remains partial. The all-dimensional proof is supplemented by 4,072 exact arithmetic regression checks. This is an unrefereed preprint; no independent review, formal verification or absolute-priority claim is asserted.