Related papers: Convex Structures Revisited

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…

We study geometric limits of convex-cocompact cyclic subgroups of the rank 1 groups SO_0(1, k+1) and SU(1, k+1). We construct examples of sequences of subgroups of such groups G that converge algebraically and whose geometric limit strictly…

We wish to attack the problems that H.~Anciaux and K.~Panagiotidou posed in [1], for non-degenerate real hypersurfaces in indefinite complex projective space. We will slightly change these authors' point of view, obtaining cleaner equations…

We associate to a real projective variety $X$ two convex cones which are fundamental in real algebraic geometry: the cone $P_X$ of quadratic forms nonnegative on $X$, and the cone $\Sigma_X$ of sums of squares of linear forms. The dual cone…

Consider a point on a convex surface in $\mathbb{R}^d$, $d \ge 2$ and a plane of support $\Pi$ to the surface at this point. Draw a plane parallel to $\Pi$ cutting a part of the surface. We study the limiting behavior of this part of…

Let $G$ be a real algebraic semi-simple Lie group and $\Gamma$ be the fundamental group of a compact negatively curved manifold. In this article we study the limit cone, introduced by Benoist, and the growth indicator function, introduced…

The rational homology of unordered configuration spaces of points on any surface was studied by Drummond-Cole and Knudsen. We compute the rational cohomology of configuration spaces on a closed orientable surface, keeping track of the mixed…

Let $L$ be a finite lattice and let $I$ be an ideal of $L$. Then the restriction map is a bounded lattice homomorphism of the congruence lattice of~$L$ into the congruence lattice of $I$. In a 2009 paper, the authors proved the converse. In…

We prove a version of the strong half-space theorem between the classes of recurrent minimal surfaces and complete minimal surfaces with bounded curvature of $\mathbb{R}^{3}_{\raisepunct{.}}$ We also show that any minimal hypersurface…

Let $A(G)$ and $B(H)$ be the Fourier and Fourier-Stieltjes algebras of locally compact groups $G$ and $H$, respectively. Ilie and Spronk have shown that continuous piecewise affine maps $\alpha: Y \subseteq H\rightarrow G$ induce completely…

In 2006, Arveson resolved a long-standing problem by showing that for any element $x$ of a separable self-adjoint unital subspace $S\subseteq B(H)$, $\|x\|=\sup\|\pi(x)\|$, where $\pi$ runs over the boundary representations for $S$. Here we…

We investigate additive properties of sets $A,$ where $A=\{a_1,a_2,\ldots ,a_k\}$ is a monotone increasing set of real numbers, and the differences of consecutive elements are all distinct. It is known that $|A+B|\geq c|A||B|^{1/2}$ for any…

We study the geometry of hyperconvex representations of hyperbolic groups in ${\rm PSL}(d,\mathbb{C})$ and establish two structural results: a group admitting a hyperconvex representation is virtually isomorphic to a Kleinian group, and its…

Let $f\colon X\to Y$ be a perfect map between finite-dimensional metrizable spaces and $p\geq 1$. It is shown that the space $C^*(X,\R^p)$ of all bounded maps from $X$ into $\R^p$ with the source limitation topology contains a dense…

Traditionally, most complex intelligence architectures are extremely non-convex, which could not be well performed by convex optimization. However, this paper decomposes complex structures into three types of nodes: operators, algorithms…

For any non-trivial convex and bounded subset $C$ of a Banach space, we show that outside of a $\sigma$-porous subset of the space of non-expansive mappings $C\to C$, all mappings have the maximal Lipschitz constant one witnessed locally at…

We show that the boundary of any bounded strongly pseudoconvex complete circular domain in $\mathbb C^2$ must contain points that are exceptionally tangent to a projective image of the unit sphere.

In this paper we introduce an enhanced notion of extremal systems for sets in locally convex topological vector spaces and obtain efficient conditions for set extremality in the convex case. Then we apply this machinery to deriving new…

Let $K$ be a convex body in an affine chart of the $n$ dimensional real Projective space $\mathbb{RP}^n$, $n \geq 3$, let $H$ be a hyperplane which is not a support hyperplane of $K$ and let $p_1,p_2 \in \mathbb{RP}^n \setminus H$ be two…

In this paper, we prove first that the iterates of a mean nonexpansive map defined on a weakly compact, convex set converge weakly to a fixed point in the presence of Opial's property and asymptotic regularity at a point. Next, we prove the…