More on logarithmic sums of convex bodies

By Christos Saroglou. 2014

60 citations recorded by the source corpus.

We prove that the log-Brunn-Minkowski inequality (log-BMI) for the Lebesque measure in dimension $n$ would imply the log-BMI and, therefore, the B-conjecture for any log-concave density in dimension $n$. As a consequence, we prove the log-BMI and the B-conjecture for any log-concave density, 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 byproduct of our methods, we study possible log-concavity of the function $t\mapsto |(K+_p\cdot e^tL)^{\circ}|$, where $p\geq 1$ 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.

Subfields: Applied Mathematics, Geometry and Topology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/93e5c231131f62327c7381b46d25e12a5546f9d5

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Citations in the corpus

Local $L^p$-Brunn-Minkowski inequalities for $p<1$ (2017, background): “• The log-Brunn–Minkowski conjecture has been shown by Saroglou [50, 51] to be intimately related to the generalized B-conjecture.”
Existence of solutions to the even dual Minkowski problem (2018, background): “The logarithmic Minkowski problem has strong connections with isotropic measures (Böröczky-LYZ [10]), curvature flows (Andrews [2, 3]), and the log-Brunn-Minkowski inequality (Böröczky-LYZ [8], Colesanti, Livshyts & Marsiglietti [15], Rotem [52], Saroglou [53]), an inequality stronger than the c”
The Minkowski problem in Gaussian probability space (2020, result): “also linked with the conjectured log-Brunn-Minkowski inequality (planar case established in [12])—an inequality in the Lebesgue measure space but with diﬀerent addition—following a result by Saroglou [62]. We also would like to mention the work of Borell [5]. Since Gaussian probability measure is no”
The dimensional Brunn–Minkowski inequality in Gauss space (2020, background): “(72) In [Sar16], Saroglou proved that the log-Brunn–Minkowski conjecture formally implies the validity of (72), with the Lebesgue measure | · | replaced by any symmetric log-concave measure μ, that is”
On the stability of Brunn–Minkowski type inequalities (2016, background): “Saroglou showed [24] that the validity of Conjecture 1.”
On the Brunn-Minkowski inequality for general measures with applications to new isoperimetric-type inequalities (2015, methodology): “Saroglou [31], generalized the inequality to the case of measures with even log-concave densities on R(2) (see Corollary 3.”
Local Lp-Brunn-Minkowski inequalities for p<1 (2017, background): “iglietti [13] veriﬁed the conjecture locally for small-enough C2-perturbations of the Euclidean ball Bn 2 (see below for more details). • The log-Brunn–Minkowski conjecture has been shown by Saroglou [58, 59] to be intimately related to the generalized B-conjecture. Further connections to a conjectu”