A conditional power stability estimate for the logarithmic Brunn-Minkowski inequality

Abstract

Assuming the smooth variance and moment-coordinate identities specified in this paper, we prove a power stability estimate for the logarithmic Brunn-Minkowski inequality on origin-symmetric convex bodies. In each dimension at least three, a small deficit forces the two bodies close in Hausdorff distance to independent positive dilations of common complementary factors. The exponent depends only on dimension; the constant also depends on fixed inner and outer radii and on an interior range of interpolation parameters. No boundary regularity, common reflection group, facet bound, or lower bound on block angles is required. The proof retains nonzero derivative energy in a compact-target moment limit, converts a weak boundary identity to an absolute flux estimate, and constructs one product body whose two dilations approximate the original endpoints. The exponent is explicit but not sharp.