Related papers: Estimates for invariant metrics on $\Bbb C$-convex…
We consider uniformly strongly elliptic systems of the second order with bounded coefficients. First, sufficient conditions for the invariance of convex bodies obtained for linear systems without zero order term in bounded domains and…
In this short note it is shown that all invariant metrics and functions of bounded $\mathcal C^2$-smooth domain coincide on an open non-empty subset.
In this paper, we study the $\bar\partial$-equation on some convex domains of infinite type in $\mathbb C^2$. In detail, we prove that supnorm estimates hold for infinite exponential type domains provided the exponent is less than 1.
We introduce two valuation-based deviations on convex bodies. Using a construction that allows us to associate to these deviations "intrinsic" pseudometrics, we establish various results which capture information about the underlying…
A classification of SL$(n)$ invariant valuations on the space of convex polytopes in $R^n$ without any continuity assumptions is established. A corresponding result is obtained on the space of convex polytopes in $R^n$ that contain the…
In this paper we obtain sharp weighted estimates for solutions of the $\partial$-equation in a lineally convex domains of finite type. Precisely we obtain estimates in spaces of the form L p ({\Omega},$\delta$ $\gamma$), $\delta$ being the…
Studying the behavior of real and complex geodesics we provide sharp estimates for the Kobayashi distance, the Lempert function, and the Carath\'eodory distance on $\mathcal{C}^{2,\alpha}$-smooth strongly pseudoconvex domains. Similar…
A $\sqrt{n}$ estimate in the hyperplane problem with arbitrary measures has recently been proved in \cite{K3}. In this note we present analogs of this result for sections of lower dimensions and in the complex case. We deduce these…
We provide a class of geometric convex domains on which the Carath\'eodory-Reiffen metric, the Bergman metric, the complete K\"ahler-Einstein metric of negative scalar curvature are uniformly equivalent, but not proportional to each other.…
In convex geometry, the constructions that assign to a convex body its difference body, projection body, or volume have the following properties: They are (1) invariant under volume-preserving linear changes of coordinates; (2) continuous;…
This article is a survey of recent results on slicing inequalities for convex bodies. The focus is on the setting of arbitrary measures in place of volume.
We show the existence of Lebesgue-equivalent conservative and ergodic $\sigma$-finite invariant measures for a wide class of one-dimensional random maps consisting of piecewise convex maps. We also estimate the size of invariant measures…
We show that domains, that allow for convex functions with unbounded gradient at their boundary, are convex.
It is shown that a lower bound of the Kobayashi metric of convex domains in C^n does not hold for non-convex domains.
In this paper we investigate the regularity properties of weighted Bergman projections for smoothly bounded pseudo-convex domains of finite type in $\mathbb{C}^{n}$. The main result is obtained for weights equal to a non negative rational…
We prove an estimate for arbitrary measure of sections of convex bodies. The proof is based on a stability result for intersection bodies.
The hermitian analog of Aleksandrov's area measures of convex bodies is investigated. A characterization of those area measures which arise as the first variation of unitarily invariant valuations is established. General smooth area…
We discuss some extremal bases for $\CC$-convex domains.
We estimate the support of a uniform density, when it is assumed to be a convex polytope or, more generally, a convex body in $\R^d$. In the polytopal case, we construct an estimator achieving a rate which does not depend on the dimension…
A discussion of methods of nonisotropic fine quantitative complex analysis on lineally convex domains of finite type is given. The needed support functions with best possible estimates are considered together with the estimation of their…