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Register a new userConditionally Evenly Convex Sets and Evenly Quasi-Convex Maps
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Evenly convex sets in a topological vector space are defined as the intersection of a family of open half spaces. We introduce a generalization of this concept in the conditional framework and provide a generalized version of the bipolar theorem. This notion is then applied to obtain the dual representation of conditionally evenly quasi-convex maps.
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We present new results on optimization problems where the involved functions are evenly convex. By means of a generalized conjugation scheme and the perturbation theory introduced by Rockafellar, we propose an alternative dual problem for a general optimization one defined on a separated locally convex topological space. Sufficient conditions for converse and total duality involving the even convexity of the perturbation function and $c$-subdifferentials are given. Formulae for the $c$-subdifferential and biconjugate of the objective function of a general optimization problem are provided, too. We also characterize the total duality also by means of the saddle-point theory for a notion of Lagrangian adapted to the considered framework.
A matrix convex set is a set of the form $mathcal{S} = cup_{ngeq 1}mathcal{S}_n$ (where each $mathcal{S}_n$ is a set of $d$-tuples of $n times n$ matrices) that is invariant under UCP maps from $M_n$ to $M_k$ and under formation of direct sums. We study the geometry of matrix convex sets and their relationship to completely positive maps and dilation theory. Key ingredients in our approach are polar duality in the sense of Effros and Winkler, matrix ranges in the sense of Arveson, and concrete constructions of scaled commuting normal dilation for tuples of self-adjoint operators, in the sense of Helton, Klep, McCullough and Schweighofer. Given two matrix convex sets $mathcal{S} = cup_{n geq 1} mathcal{S}_n,$ and $mathcal{T} = cup_{n geq 1} mathcal{T}_n$, we find geometric conditions on $mathcal{S}$ or on $mathcal{T}$, such that $mathcal{S}_1 subseteq mathcal{T}_1$ implies that $mathcal{S} subseteq Cmathcal{S}$ for some constant $C$. For instance, under various symmetry conditions on $mathcal{S}$, we can show that $C$ above can be chosen to equal $d$, the number of variables, and in some cases this is sharp. We also find an essentially unique self-dual matrix convex set $mathcal{D}$, the self-dual matrix ball, for which corresponding inclusion and dilation results hold with constant $C=sqrt{d}$. Our results have immediate implications to spectrahedral inclusion problems studied recently by Helton, Klep, McCullough and Schweighofer. Our constants do not depend on the ranks of the pencils determining the free spectrahedra in question, but rather on the number of variables $d$. There are also implications to the problem of existence of (unital) completely positive maps with prescribed values on a set of operators.
For a finite point set in $mathbb{R}^d$, we consider a peeling process where the vertices of the convex hull are removed at each step. The layer number $L(X)$ of a given point set $X$ is defined as the number of steps of the peeling process in order to delete all points in $X$. It is known that if $X$ is a set of random points in $mathbb{R}^d$, then the expectation of $L(X)$ is $Theta(|X|^{2/(d+1)})$, and recently it was shown that if $X$ is a point set of the square grid on the plane, then $L(X)=Theta(|X|^{2/3})$. In this paper, we investigate the layer number of $alpha$-evenly distributed point sets for $alpha>1$; these point sets share the regularity aspect of random point sets but in a more general setting. The set of lattice points is also an $alpha$-evenly distributed point set for some $alpha>1$. We find an upper bound of $O(|X|^{3/4})$ for the layer number of an $alpha$-evenly distributed point set $X$ in a unit disk on the plane for some $alpha>1$, and provide an explicit construction that shows the growth rate of this upper bound cannot be improved. In addition, we give an upper bound of $O(|X|^{frac{d+1}{2d}})$ for the layer number of an $alpha$-evenly distributed point set $X$ in a unit ball in $mathbb{R}^d$ for some $alpha>1$ and $dgeq 3$.
We consider the convex hull of the perturbed point process comprised of $n$ i.i.d. points, each distributed as the sum of a uniform point on the unit sphere $S^{d-1}$ and a uniform point in the $d$-dimensional ball centered at the origin and of radius $n^{alpha}, alpha in (-infty, infty)$. This model, inspired by the smoothed complexity analysis introduced in computational geometry cite{DGGT,ST}, is a perturbation of the classical random polytope. We show that the perturbed point process, after rescaling, converges in the scaling limit to one of five Poisson point processes according to whether $alpha$ belongs to one of five regimes. The intensity measure of the limit Poisson point process undergoes a transition at the values $alpha = frac{-2} {d -1}$ and $alpha = frac{2} {d + 1}$ and it gives rise to four rescalings for the $k$-face functional on perturbed data. These rescalings are used to establish explicit expectation asymptotics for the number of $k$-dimensional faces of the convex hull of either perturbed binomial or Poisson data. In the case of Poisson input, we establish explicit variance asymptotics and a central limit theorem for the number of $k$-dimensional faces. Finally it is shown that the rescaled boundary of the convex hull of the perturbed point process converges to the boundary of a parabolic hull process.
The notion of a valuation on convex bodies is very classical. The notion of a valuation on a class of functions was recently introduced and studied by M. Ludwig and others. We study an explicit relation between continuous valuations on convex functions which are invariant under adding arbitrary linear functionals, and translations invariant continuous valuations on convex bodies. More precisely, we construct a natural linear map from the former space to the latter and prove that it has dense image and infinite dimensional kernel. The proof uses the authors irreducibility theorem and few properties of the real Monge-Ampere operators due to A.D. Alexandrov and Z. Blocki. Fur- thermore we show how to use complex, quaternionic, and octonionic Monge-Ampere operators to construct more examples of continuous valuations on convex functions in an analogous way.
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