Related papers: Volume product
The $L_p$ versions of the support function and polar body are introduced by Berndtsson, Mastrantonis and Rubinstein in \cite{Berndtsson-Mastrantonis-Rubinstein-2023} recently. In this paper, we prove that the $L_p$-support function of the…
In the present paper, the following convexity principle is proved: any closed convex multifunction, which is metrically regular in a certain uniform sense near a given point, carries small balls centered at that point to convex sets, even…
In this Letter, we have examined the thermodynamic volume products for spherically symmetric and axisymmetric spacetimes in the framework of \emph{extended phase space}. Such volume products usually formulated in terms of the outer…
We use symplectic techniques to obtain partial results on Mahler's conjecture about the product of the volume of a convex body and the volume of its polar. We confirm the conjecture for hyperplane sections or projections of $\ell_p$-balls…
Coverings of convex bodies have emerged as a central component in the design of efficient solutions to approximation problems involving convex bodies. Intuitively, given a convex body $K$ and $\epsilon> 0$, a covering is a collection of…
It is well known that the convex hull of $\{(x,y,xy)\}$, where $(x,y)$ is constrained to lie in a box, is given by the Reformulation-Linearization Technique (RLT) constraints. Belotti {\em et al.\,}(2010) and Miller {\em et al.\,}(2011)…
One of the most fruitful results from Minkowski's geometric viewpoint on number theory is his so called 1st Fundamental Theorem. It provides an optimal upper bound for the volume of an o-symmetric convex body whose only interior lattice…
We develop a general comparison result for inf convolution operators related to rearrangement. As a consequence we derive comparison results under spherically symmetric rearrangement for Laplace and polar transforms. As a by product we…
We study numerical restricted volumes of (1,1) classes on compact Kahler manifolds, as introduced by Boucksom. Inspired by work of Ein-Lazarsfeld-Mustata-Nakamaye-Popa on restricted volumes of line bundles on projective manifolds, we pose a…
Bohn, Faenza, Fiorini, Fisikopoulos, Macchia, and Pashkovich (2015) conjectured that 2-level polytopes cannot simultaneously have many vertices and many facets, namely, that the maximum of the product of the number of vertices and facets is…
Take a set of balls in $\mathbb R^d$. We find a subset of pairwise disjoint balls whose combined perimeter controls the perimeter of the union of the original balls. This can be seen as a boundary version of the Vitali covering lemma. We…
Motivated by a long-standing conjecture of Polya and Szeg\"o about the Newtonian capacity of convex bodies, we discuss the role of concavity inequalities in shape optimization, and we provide several counterexamples to the…
We derive lower bounds in best rational approximation of given degree to finite Blaschke products, in the Hardy space $H^2$ of the unit disk. We first consider approximation to $z^N$, and then move on to more general Blaschke products whose…
We investigate properties of the pseudo-Riemannian volume, entropy, and diameter for convex cocompact representations $\rho : \Gamma \to \mathrm{SO}(p,q+1)$ of closed $p$-manifold groups. In particular: We provide a uniform lower bound of…
We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space…
In this note, we estimate the upper bound of volume of closed positively or nonnegatively curved Alexandrov space $X$ with strictly convex boundary. We also discuss the equality case. In particular, the Boundary Conjecture holds when the…
We provide a description of the space of continuous and translation invariant Minkowski valuations $\Phi:\mathcal{K}^n\to\mathcal{K}^n$ for which there is an upper and a lower bound for the volume of $\Phi(K)$ in terms of the volume of the…
For $p>1$, we derive explicit formulas for the intrinsic volumes $V_0(\mathbb B_p^n),\dots,V_{n-1}(\mathbb B_p^n)$ of the $n$-dimensional $\ell_p$-balls $$ \mathbb B_p^n = \{x\in\mathbb R^n:\ |x_1|^p+\ldots+|x_n|^p\le 1\} $$ and, more…
We define a scale of L^q Carleson norms, all of which characterize the membership of a function in BMO. The phenomenon is analogous to the John-Nirenberg inequality, but on the level of Carleson measures. The classical Carleson condition…
It is proved that if $u_1,\ldots, u_n$ are vectors in ${\Bbb R}^k, k\le n, 1 \le p < \infty$ and $$r = ({1\over k} \sum ^n_1 |u_i|^p)^{1\over p}$$ then the volume of the symmetric convex body whose boundary functionals are $\pm u_1,\ldots,…