Logarithmic and Lp Brunn–Minkowski inequalities and the B-conjecture. Proves the logarithmic Brunn–Minkowski inequality for origin-symmetric convex bodies in every dimension, and the scalar-dilation B-conjecture for all even log-concave Radon measures. For Lebesgue volume it also proves the additive Lp Brunn–Minkowski inequality for full-dimensional origin-symmetric convex bodies throughout $0\lt p\lt 1$.
released 2026-09-23 | 1 theorem · 3 lemmas · 7 proofs · 7,409 words |
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We prove the logarithmic Brunn–Minkowski conjecture for arbitrary origin-symmetric convex bodies in every dimension. The theorem also gives the symmetric Lp Brunn–Minkowski inequality for every $0\lt p\lt 1$. Combined with Saroglou's transfer theorem and a support-subspace reduction, it yields the logarithmic inequality for every even log-concave Radon measure and the scalar-dilation $(B)$-conjecture.