Related papers: Convex defining functions for convex domains
We show that a closed piecewise-linear hypersurface immersed in $R^n$ ($n\ge 3$) is the boundary of a convex body if and only if every point in the interior of each $(n-3)$-face has a neighborhood that lies on the boundary of some convex…
The Hessian of a differentiable convex function is positive semidefinite. Therefore, checking the Hessian of a given function is a natural approach to certify convexity. However, implementing this approach is not straightforward since it…
We prove some results concerning the boundary of a convex set in $\H^n$. This includes the convergence of curvature measures under Hausdorff convergence of the sets, the study of normal points, and, for convex surfaces, a generalized Gauss…
In the present paper, the order of convexity of z\Gauss(a,b;c;z) is first given under some conditions on the positive real parameters a, b and c. Then we show that the image domains of the unit disc \D under some shifted zero-balanced…
We show that every bounded pseudoconvex domain with H\"older boundary in $\mathbb C^n$ is hyperconvex.
We consider Laplacian eigenfunctions in circular, spherical and elliptical domains in order to discuss three kinds of high-frequency localization: whispering gallery modes, bouncing ball modes, and focusing modes. Although the existence of…
Functions that are holomorphic and Lipschitz in a smoothly bounded domain enjoy a gain in the order of Lipschitz regularity in the complex tangential directions near the boundary. We describe this gain explicitly in terms of the defining…
This note deals with certain properties of convex functions. We provide results on the convexity of the set of minima of these functions, the behaviour of their subgradient set under restriction, and optimization of these functions over an…
The objective of this paper is to characterize harmonic Hardy spaces and a boundary behavior of harmonic functions on a smooth domain in real Euclidean space.
We consider a space of infinitely smooth functions on an unbounded closed convex set in ${\mathbb R}^n$. It is shown that each function of this space can be extended to an entire function in ${\mathbb C}^n$ satisfying some prescribed growth…
We show how an operation of inf-convolution can be used to approximate convex functions with $C^{1}$ smooth convex functions on Riemannian manifolds with nonpositive curvature (in a manner that not only is explicit but also preserves some…
We consider Bergman spaces and variations of them in one or several complex variables. For some domains we show that in these spaces the generic function is totally unbounded and hence non - extendable. We also show that the generic…
We establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded…
For any sequence of properly convex domains in the real projective plane such that the zeros of Pick differentials have bounded multiplicity and get further and further apart, we determined all Hausdorff limit domains that one can obtain…
We give an explicit verifiable characterization of weakly pseudoconvex but locally nonconvexifiable hypersurfaces of finite type in dimension two. It is expressed in terms of a generalized model, which captures local geometry of the…
By considering a certain univalent function in the open unit disk U, that maps U onto a strip domain, we introduce a new class of analytic and close-to-convex functions by means of a certain non-homogeneous Cauchy-Euler-type differential…
We show that in $\mathbb{C}^2$ if the set of strongly regular points are closed in the boundary of a smooth bounded pseudoconvex domain, then the domain is c-regular, that is, the plurisubharmonic upper envelopes of functions continuous up…
Given a bounded strongly pseudoconvex domain $D$ in $\mathbb{C}^n$ with smooth boundary, we give a characterization through products of functions in weighted Bergman spaces of $(\lambda,\gamma)$-skew Carleson measures on $D$, with…
This survey is devoted to discussing the problems of the unique determination of surfaces that are the boundaries of (generally speaking) nonconvex domains. First (in Sec. 2) we examine some results on the problem of the unique…
In this paper we study the automorphism group of smoothly bounded convex domains. We show that such a domain is biholomorphic to a "polynomial ellipsoid" (that is, a domain defined by a weighted homogeneous balanced polynomial) if and only…