The Mahler conjecture for general (non-symmetric) convex bodies

Theorem 1.1: for every $n\ge1$ and every convex body $K\subset\mathbb R^n$ with Santalo point $s(K)$, $|K|\,|(K-s(K))^\circ|\ge(n+1)^{n+1}/(n!)^2$, with equality iff $K$ is a simplex; no symmetry or boundary regularity is assumed. The proof goes through Klartag's cone and Laplace-transform correspondence, Gaussian-averaged nearest-point projections onto a cone and its dual, and a spectral comparison whose scalar estimates rest on finite rational certificates with a released Python runner. Corollary: the sharp functional inequality $\int e^{-\varphi}\int e^{-\varphi^*}\ge e^n$ for convex $\varphi$, via Fradelizi-Meyer, and an entropy-transport bound under an extra hypothesis. It does not give the symmetric constant $4^n/n!$, which is a separate entry.

Theorem 1.1: for every $n\ge1$ and every convex body $K\subset\mathbb R^n$ with Santalo point $s(K)$, $|K|\,|(K-s(K))^\circ|\ge(n+1)^{n+1}/(n!)^2$, with equality iff $K$ is a simplex; no symmetry or boundary regularity is assumed. The proof goes through Klartag's cone and Laplace-transform correspondence, Gaussian-averaged nearest-point projections onto a cone and its dual, and a spectral comparison whose scalar estimates rest on finite rational certificates with a released Python runner. Corollary: the sharp functional inequality $\int e^{-\varphi}\int e^{-\varphi^*}\ge e^n$ for convex $\varphi$, via Fradelizi-Meyer, and an entropy-transport bound under an extra hypothesis. It does not give the symmetric constant $4^n/n!$, which is a separate entry.