The logarithmic Brunn-Minkowski conjecture for origin-symmetric convex bodies

Theorem 1.1: for every $n\ge1$, origin-symmetric convex bodies $K,L$ and $0\le\lambda\le1$, $|W[h_K^{1-\lambda}h_L^\lambda]|\ge|K|^{1-\lambda}|L|^\lambda$. Corollary 1.2 gives the symmetric $L_p$ Brunn-Minkowski inequality for full-dimensional origin-symmetric bodies and every $0<p<1$, via the monotone implication recorded by Boroczky-Lutwak-Yang-Zhang. The proof reduces to a variance bound for sums of even powers of linear forms, proved in moment-measure coordinates. It does not treat equality cases, non-symmetric bodies (where the inequality is false in general), or the uniqueness question for the even logarithmic Minkowski problem.

Theorem 1.1: for every $n\ge1$, origin-symmetric convex bodies $K,L$ and $0\le\lambda\le1$, $|W[h_K^{1-\lambda}h_L^\lambda]|\ge|K|^{1-\lambda}|L|^\lambda$. Corollary 1.2 gives the symmetric $L_p$ Brunn-Minkowski inequality for full-dimensional origin-symmetric bodies and every $0<p<1$, via the monotone implication recorded by Boroczky-Lutwak-Yang-Zhang. The proof reduces to a variance bound for sums of even powers of linear forms, proved in moment-measure coordinates. It does not treat equality cases, non-symmetric bodies (where the inequality is false in general), or the uniqueness question for the even logarithmic Minkowski problem.