Related papers: On angles between convex sets in Hilbert spaces

We introduce the convex bundle method to solve convex, non-smooth optimization problems on Riemannian manifolds of bounded sectional curvature. Each step of our method is based on a model that involves the convex hull of previously…

We prove that, any problem of minimization of proper lower semicontinuous function defined on a normal Hausdorff space, is canonically equivalent to a problem of minimization of a proper weak * lower semicontinuous convex function defined…

We study two notions. One is that of spindle convexity. A set of circumradius not greater than one is spindle convex if, for any pair of its points, it contains every short circular arc of radius at least one, connecting them. The other…

The problem of finding the minimizer of a sum of convex functions is central to the field of distributed optimization. Thus, it is of interest to understand how that minimizer is related to the properties of the individual functions in the…

Consider a convex domain B of space. We prove that there exist complete minimal surfaces which are properly immersed in B. We also demonstrate that if D and D' are convex domains with D bounded and the closure of D contained in D' then any…

This paper explores a natural generalization of Euclidean projection through the lens of strongly quasiconvex functions, as developed in prior works. By establishing a connection between strongly quasiconvex functions and the theory of…

Divergences are quantities that measure discrepancy between two probability distributions and play an important role in various fields such as statistics and machine learning. Divergences are non-negative and are equal to zero if and only…

According to a result due to B.T. Polyak, a mapping between Hilbert spaces, which is $C^{1,1}$ around a regular point, carries a ball centered at that point to a convex set, provided that the radius of the ball is small enough. The present…

We describe the minimax reconstruction rates in linear ill-posed equations in Hilbert space when smoothness is given in terms of general source sets. The underlying fundamental result, the minimax rate on ellipsoids, is proved similarly to…

We establish some relations between the perimeter, the area and the visual angle of a planar compact convex set. Our first result states that Crofton's formula is the unique universal formula relating the visual angle, length and area.…

A theorem of Tietze and Nakamija, from 1928, asserts that if a subset X of R^n is closed, connected, and locally convex, then it is convex. We give an analogous "local to global convexity" theorem when the inclusion map of X to R^n is…

We construct geometric barriers for minimal graphs in H^n xR. We prove the existence and uniqueness of a solution of the vertical minimal equation in the interior of a convex polyhedron in H^n extending continuously to the interior of each…

A set in the Euclidean plane is said to be biconvex if, for some angle $\theta\in[0,\pi/2)$, all its sections along straight lines with inclination angles $\theta$ and $\theta+\pi/2$ are convex sets (i.e, empty sets or segments).…

Let $\Omega\subset\mathbb{R}^{n+1}$ have minimal Gaussian surface area among all sets satisfying $\Omega=-\Omega$ with fixed Gaussian volume. Let $A=A_{x}$ be the second fundamental form of $\partial\Omega$ at $x$, i.e. $A$ is the matrix of…

Let $P$ be a planar set of $n$ points in general position. We consider the problem of computing an orientation of the plane for which the Rectilinear Convex Hull of $P$ has minimum area. Bae et al. (Computational Geometry: Theory and…

We consider an extension of the conventional quantum Heisenberg algebra, assuming that coordinates as well as momenta fulfil nontrivial commutation relations. As a consequence, a minimal length and a minimal mass scale are implemented. Our…

Given a set of disjoint simple polygons $\sigma_1, \ldots, \sigma_n$, of total complexity $N$, consider a convexification process that repeatedly replaces a polygon by its convex hull, and any two (by now convex) polygons that intersect by…

Convex geometries form a subclass of closure systems with unique criticals, or $UC$-systems. We show that the $F$-basis introduced in [1] for $UC$-systems, becomes optimum in convex geometries, in two essential parts of the basis: right…

Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in $H^k(\Omega)$, and optimizing functionals arising from some…

In this paper the theory of uniformly convex metric spaces is developed. These spaces exhibit a generalized convexity of the metric from a fixed point. Using a (nearly) uniform convexity property a simple proof of reflexivity is presented…