Related papers: Extreme and Exposed Points Arising from Rational K…

We prove an upper bound for the number of rational points of bounded height on irreducible affine hypersurfaces. More precisely, given an irreducible polynomial $f \in \mathbb{Z}[X_1, \dots, X_n]$, we prove an upper bound on the number of…

Let $\Delta$ be the Dunkl Laplacian on $\mathbb R^N$ associated with a normalized root system $R$ and a multiplicity function $k(\alpha)\geq 0$. We say that a function $f$ belongs to the Hardy space $H^1_{\Delta}$ if the nontangential…

The variation distance closure of an exponential family with a convex set of canonical parameters is described, assuming no regularity conditions. The tools are the concepts of convex core of a measure and extension of an exponential…

We solve a randomized version of the following open question: is there a strictly convex, bounded curve \gamma in the plane such that the number of rational points on \gamma, with denominator $n$, approaches infinity with $n$? Although this…

In this paper we introduce and study the concept of set extremality for systems of convex sets in vector spaces without topological structures. Characterizations of the extremal systems of sets are obtained in the form of the convex…

Convex clustering is a well-regarded clustering method, resembling the similar centroid-based approach of Lloyd's $k$-means, without requiring a predefined cluster count. It starts with each data point as its centroid and iteratively merges…

This article studies generalizations of (matrix) convexity, including partial convexity and biconvexity, under the umbrella of $\Gamma$-convexity. Here $\Gamma$ is a tuple of free symmetric polynomials determining the geometry of a…

We construct a continuum of non-homeomorphic compact subspaces of the real line R without singleton components. Thus from the purely topological point of view the real line contains not only more closed sets than open sets but also more…

In this article a class of closed convex sets in the Euclidean $n$-space which are the convex hull of their profiles is described. Thus a generalization of Krein-Milman theorem\cite{Lay:1982} to a class of closed non-compact convex sets is…

We prove a weak fundamental principle for $\lambda$-homogeneous solutions of homogeneous constant-coefficient systems on open pointed convex cones. Starting with the solution family $S_{\mathcal B}$ arising in the Ehrenpreis--Palamodov…

In this paper we consider the unbounded local completely positive and local completely contractive maps on maximal tensor product of unital locally C*-algebras and discuss on extremal points of certain convex subsets in the set of such…

In a normed linear space X an element x is said to be orthogonal to another element y in the sense of Birkhoff-James, written as $ x \perp_{B}y, $ iff $ \| x \| \leq \| x + \lambda y \| $ for all scalars $ \lambda.$ We prove that a normed…

Let $C$ be a convex subset of a locally convex space. We provide optimal approximate fixed point results for sequentially continuous maps $f\colon C\to\bar{C}$. First we prove that if $f(C)$ is totally bounded, then it has an approximate…

Let C be a real nonsingular affine curve of genus one, embedded in affine n-space, whose set of real points is compact. For any polynomial f which is nonnegative on C(R), we prove that there exist polynomials f_i with f \equiv \sum_i f_i^2…

Let $P$ be a set of $n$ points in general position on the plane. A set of closed convex polygons with vertices in $P$, and with pairwise disjoint interiors is called a convex decomposition of $P$ if their union is the convex hull of $P$,…

We construct a weakly compact convex subset of $\ell^2$ with nonempty interior that has an isolated maximal element, with respect to the lattice order $\ell _+^2$. Moreover, the maximal point cannot be supported by any strictly positive…

It is a classical result of Mahler that for any rational number $\alpha$ > 1 which is not an integer and any real 0 < c < 1, the set of positive integers n such that $\alpha$ n < c n is necessarily finite. Here for any real x, x denotes the…

We characterize the (essentially) decreasing sequences of positive numbers $\beta$ = ($\beta$ n) for which all composition operators on H 2 ($\beta$) are bounded, where H 2 ($\beta$) is the space of analytic functions f in the unit disk…

Let X be a projective cubic hypersurface of dimension 11 or more, which is defined over the rationals. In this paper it is shown that X contains rational points provided that the cubic form defining X can be written as the sum of two forms…

The Hardy space H^2(R) for the upper half plane together with a unimodular function group representation u(\lambda) = \exp(i(\lambda_1\psi_1 + ... + \lambda_n\psi_n)) for \lambda in R^n, gives rise to a manifold M of orthogonal projections…