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MORE ON LOGARITHMIC SUMS OF CONVEX BODIES

  • Christos Saroglou (a1)

Abstract

We prove that the log-Brunn–Minkowski inequality (log-BMI) for the Lebesgue measure in dimension $n$ would imply the log-BMI and, therefore, the B-conjecture for any even log-concave measure in dimension $n$ . As a consequence, we prove the log-BMI and the B-conjecture for any even log-concave measure in the plane. Moreover, we prove that the log-BMI reduces to the following: for each dimension $n$ , there is a density $f_{n}$ , which satisfies an integrability assumption, so that the log-BMI holds for parallelepipeds with parallel facets, for the density $f_{n}$ . As a byproduct of our methods, we study possible log-concavity of the function $t\mapsto |(K+_{p}\cdot ~\text{e}^{t}L)^{\circ }|$ , where $p\geqslant 1$ and $K$ , $L$ are symmetric convex bodies, which we are able to prove in some instances and, as a further application, we confirm the variance conjecture in a special class of convex bodies. Finally, we establish a non-trivial dual form of the log-BMI.

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