Related papers: The Reverse-log-Brunn-Minkowski inequality
\footnotesize B\"{o}r\"{o}czky, Lutwak, Yang and Zhang recently conjectured a certain strengthening of the Brunn-Minkowski inequality for symmetric convex bodies, the so-called log-Brunn-Minkowski inequality. We establish this inequality…
B\"or\"oczky, Lutwak, Yang and Zhang recently proved the log-Brunn-Minkowski inequality which is stronger than the classical Brunn-Minkowski inequality for two origin-symmetric convex bodies in the plane. This paper establishes the…
Log-Brunn-Minkowski inequality was conjectured by Bor\"oczky, Lutwak, Yang and Zhang \cite{BLYZ}, and it states that a certain strengthening of the classical Brunn-Minkowski inequality is admissible in the case of symmetric convex sets. It…
We prove the log-Brunn-Minkowski conjecture for convex bodies with symmetries to $n$ independent hyperplanes, and discuss the equality case and the uniqueness of the solution of the related case of the logarithmic Minkowski problem. We also…
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…
In this short note we explain why the log-Brunn-Minkowski conjecture is correct for complex convex bodies. We do this by relating the conjecture to the notion of complex interpolation, and appealing to a general theorem by…
We prove that the log-Brunn-Minkowski inequality \begin{equation*} |\lambda K+_0 (1-\lambda)L|\geq |K|^{\lambda}|L|^{1-\lambda} \end{equation*} (where $|\cdot|$ is the Lebesgue measure and $+_0$ is the so-called log-addition) holds when…
We study several of the recent conjectures in regards to the role of symmetry in the inequalities of Brunn-Minkowski type, such as the $L_p$-Brunn-Minkowski conjecture of B\"or\"oczky, Lutwak, Yang and Zhang, and the Dimensional…
We show that for any log-concave measure $\mu$ on $\mathbb{R}^n$, any pair of symmetric convex sets $K$ and $L$, and any $\lambda\in [0,1],$ $$\mu((1-\lambda) K+\lambda L)^{c_n}\geq (1-\lambda) \mu(K)^{c_n}+\lambda\mu(L)^{c_n},$$ where…
The theory of coconvex bodies was formalized by A.~Khovanski{\u\i} and V.~Timorin in \cite{KT}. It has fascinating relations with the classical theory of convex bodies, as well as applications to Lorentzian geometry. In a recent preprint…
The Lp-Brunn-Minkowski inequality palys a central role in the Brunn-Minkowski theory proposed by Firey [13] in 60's and developed by Lutwak [26,27] in 90's, which generalizes the classical Brunn-Minkowski inequality by Lp-sum of convex…
We develop a reverse entropy power inequality for convex measures, which may be seen as an affine-geometric inverse of the entropy power inequality of Shannon and Stam. The specialization of this inequality to log-concave measures may be…
We prove that for any semi-norm $\|\cdot\|$ on $\mathbb{R}^n,$ and any symmetric convex body $K$ in $\mathbb{R}^n,$ \begin{equation}\label{ineq-abs2} \int_{\partial K} \frac{\|n_x\|^2}{\langle x,n_x\rangle}\leq…
This paper establishes two new geometric inequalities in the dual Brunn-Minkowski theory. The first, originally conjectured by Lutwak, is the Brunn-Minkowski inequality for dual quermassintegrals of origin-symmetric convex bodies. The…
In this paper we present new versions of the classical Brunn-Minkowski inequality for different classes of measures and sets. We show that the inequality \[ \mu(\lambda A + (1-\lambda)B)^{1/n} \geq \lambda \mu(A)^{1/n} +…
In an earlier paper \cite{mazeng} the authors introduced the notion of curvature entropy, and proved the plane log-Minkowski inequality of curvature entropy under the symmetry assumption. In this paper we demonstrate the plane log-Minkowski…
The $L^p$-Brunn-Minkowski theory for $p\geq 1$, proposed by Firey and developed by Lutwak in the 90's, replaces the Minkowski addition of convex sets by its $L^p$ counterpart, in which the support functions are added in $L^p$-norm.…
We consider a different $L^p$-Minkowski combination of compact sets in $\mathbb{R}^n$ than the one introduced by Firey and we prove an $L^p$-Brunn-Minkowski inequality, $p \in [0,1]$, for a general class of measures called convex measures…
By studying $L^p$-combinations of strongly isomorphic polytopes, we prove the equivalence of the $L^p$-Brunn-Minkowski inequality conjectured by B\"or\"oczky, Lutwak, Yang and Zhang to the local version of the inequality studied by…
A generalization of Young's inequality for convolution with sharp constant is conjectured for scenarios where more than two functions are being convolved, and it is proven for certain parameter ranges. The conjecture would provide a unified…