Related papers: Interpolations, convexity and geometric inequaliti…
The Brunn-Minkowski theory in convex geometry concerns, among other things, the volumes, mixed volumes, and surface area measures of convex bodies. We study generalizations of these concepts to Borel measures with density in…
We study multiple sampling and interpolation problems with unbounded multiplicities in the weighted Bergman space, both in the hilbertian case p = 2 and the uniform case p = +$\infty$.
We obtain inequalities of H\"{o}lder and Minkowski type with weights generalizing both the case of weights with alternating signs and the classical case of non-negative weights.
We consider a deformation of Segal-Bargmann space and its transform. We study L^p properties of this transform and obtain entropy-entropy inequalities (Hirschman) and entropy-energy inequalities (log-Sobolev) that generalize the…
We study the connection between the concavity properties of a measure $\nu$ and the convexity properties of the associated relative entropy $D(\cdot \Vert \nu)$ along optimal transport. As a corollary we prove a new dimensional…
We prove a new family of inequalities, which compare the integral of a geometric convolution of non-negative functions with the integrals of the original functions. For classical inf-convolution, this type of inequality is called the…
We carry out calculations of Orlicz cohomology for some basic Riemannian manifolds (the real line, the hyperbolic plane, the ball). Relationship between Orlicz cohomology and Poincar\'e--Sobolev--Orlicz-type inequalities is discussed.
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…
This paper describes the theory of Minkowski problems for geometric measures in convex geometric analysis. The theory goes back to Minkowski and Aleksandrov and has been developed extensively in recent years. The paper surveys classical and…
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 survey elementary results in Minkowski spaces (i.e. finite dimensional Banach spaces) that deserve to be collected together, and give simple proofs for some of them. We place special emphasis on planar results. Many of these results have…
We study a general metric constrained interpolation problem in a de Branges-Rovnyak space $\mathcal{H}(K_S)$ associated with a contractive multiplier $S$ between two Fock spaces along with its commutative counterpart, a de Branges-Rovnyak…
Eigenvalues inequalities involving (log) convex/concav functions and Hermitian matrices, positive unital maps are considered. Simple proofs of Bhatia-Kittaneh inequality and Naimark dilation theorem are given.
A quantitative version of Minkowski sum, extending the definition of $\theta$-convolution of convex bodies, is studied to obtain extensions of the Brunn-Minkowski and Zhang inequalities, as well as, other interesting properties on Convex…
The Ehrhard-Borell inequality is a far-reaching refinement of the classical Brunn-Minkowski inequality that captures the sharp convexity and isoperimetric properties of Gaussian measures. Unlike in the classical Brunn-Minkowski theory, the…
In this paper we study how certain symmetries of convex bodies affect their geometric properties. In particular, we consider the impact of symmetries generated by the block diagonal subgroup of orthogonal transformations, generalizing…
We deal with Monge-Amp\`ere type equations modeled upon general anisotropic norms $H$ in $\mathbb R^n$. An overdetermined problem for convex solutions to these equations is analyzed. The relevant solutions are subject to both a homogeneous…
Let $u$ be a smooth, strictly $k$-plurisubharmonic function on a bounded domain $\Omega\in\mathbb C^n$ with $2\leq k\leq n$. The purpose of this paper is to study the regularity of solution to the linearized complex Monge-Amp\`ere and…
We prove an inequality for unitarily invariant norms that interpolates between the Arithmetic-Geometric Mean inequality and the Cauchy-Schwarz inequality.
We combine functional analytic and geometric viewpoints on approximate Birkhoff and isosceles orthogonality in generalized Minkowski spaces which are finite-dimensional vector spaces equipped with a gauge. This is the first approach to…