An isomorphic version of the Busemann-Petty problem for arbitrary measures

We prove the following theorem. Let $\mu$ be a measure on $R^n$ with even continuous density, and let $K,L$ be origin-symmetric convex bodies in $R^n$ so that $\mu(K\cap H)\le \mu(L\cap H)$ for any central hyperplane H. Then $\mu(K)\le \sqrt{n} \mu(L).$ We also prove this result with better constants for some special classes of measures and bodies. Finally, we prove a version of the hyperplane inequality for convex measures

Bezout Inequality for Mixed volumes

In this paper we consider the following analog of Bezout inequality for mixed volumes: $$V(P_1,\dots,P_r,\Delta^{n-r})V_n(\Delta)^{r-1}\leq \prod_{i=1}^r V(P_i,\Delta^{n-1})\ \text{ for }2\leq r\leq n.$$ We show that the above inequality is true when $\Delta$ is an $n$-dimensional simplex and $P_1, \dots, P_r$ are convex bodies in $\mathbb{R}^n$. We conjecture that if the above inequality is true for all convex bodies $P_1, \dots, P_r$, then $\Delta$ must be an $n$-dimensional simplex. We prove that if the above inequality is true for all convex bodies $P_1, \dots, P_r$, then $\Delta$ must be indecomposable (i.e. cannot be written as the Minkowski sum of two convex bodies which are not homothetic to $\Delta$), which confirms the conjecture when $\Delta$ is a simple polytope and in the 2-dimensional case. Finally, we connect the inequality to an inequality on the volume of orthogonal projections of convex bodies as well as prove an isomorphic version of the inequality.Comment: 18 pages, 2 figures; an error in the isomorphic version of the inequality is corrected (which improved the inequality

Wulff shapes and a characterization of simplices via a Bezout type inequality

Inspired by a fundamental theorem of Bernstein, Kushnirenko, and Khovanskii we study the following Bezout type inequality for mixed volumes $$V(L_1,\dots,L_{n})V_n(K)\leq V(L_1,K[{n-1}])V(L_2,\dots, L_{n},K).$$ We show that the above inequality characterizes simplices, i.e. if $K$ is a convex body satisfying the inequality for all convex bodies $L_1, \dots, L_n \subset {\mathbb R}^n$, then $K$ must be an $n$-dimensional simplex. The main idea of the proof is to study perturbations given by Wulff shapes. In particular, we prove a new theorem on differentiability of the support function of the Wulff shape, which is of independent interest. In addition, we study the Bezout inequality for mixed volumes introduced in arXiv:1507.00765 . We introduce the class of weakly decomposable convex bodies which is strictly larger than the set of all polytopes that are non-simplices. We show that the Bezout inequality in arXiv:1507.00765 characterizes weakly indecomposable convex bodies

The geometry of p-convex intersection bodies

Busemann's theorem states that the intersection body of an origin-symmetric convex body is also convex. In this paper we provide a version of Busemann's theorem for p-convex bodies. We show that the intersection body of a p-convex body is q-convex for certain q. Furthermore, we discuss the sharpness of the previous result by constructing an appropriate example. This example is also used to show that IK, the intersection body of K, can be much farther away from the Euclidean ball than K. Finally, we extend these theorems to some general measure spaces with log-concave and $s$-concave measure

Characterization of Simplices via the Bezout Inequality for Mixed volumes

We consider the following Bezout inequality for mixed volumes: $$V(K_1,\dots,K_r,\Delta[{n-r}])V_n(\Delta)^{r-1}\leq \prod_{i=1}^r V(K_i,\Delta[{n-1}])\ \text{ for }2\leq r\leq n.$$ It was shown previously that the inequality is true for any $n$-dimensional simplex $\Delta$ and any convex bodies $K_1, \dots, K_r$ in $\mathbb{R}^n$. It was conjectured that simplices are the only convex bodies for which the inequality holds for arbitrary bodies $K_1, \dots, K_r$ in $\mathbb{R}^n$. In this paper we prove that this is indeed the case if we assume that $\Delta$ is a convex polytope. Thus the Bezout inequality characterizes simplices in the class of convex $n$-polytopes. In addition, we show that if a body $\Delta$ satisfies the Bezout inequality for all bodies $K_1, \dots, K_r$ then the boundary of $\Delta$ cannot have strict points. In particular, it cannot have points with positive Gaussian curvature.Comment: 8 page