The number of weakly compact convex subsets of the Hilbert space

Uniformly convex subsets of the Hilbert space with modulus of convexity of the second order

We prove that in the Hilbert space every uniformly convex set with modulus of convexity of the second order at zero is an intersection of closed balls of fixed radius. We also obtain an estimate of this radius.

Weakly convex sets and modulus of nonconvexity

We consider a definition of a weakly convex set which is a generalization of the notion of a weakly convex set in the sense of Vial and a proximally smooth set in the sense of Clarke, from the case of the Hilbert space to a class of Banach spaces with the modulus of convexity of the second order. Using the new definition of the weakly convex set with the given modulus of nonconvexity we prove a new retraction theorem and we obtain new results about continuity of the intersection of two continuous set-valued mappings (one of which has nonconvex images) and new affirmative solutions of the split…

关于弱 $1$-凸集与弱 $1$-半凸集

On weakly $1$-convex and weakly $1$-semiconvex sets

本文定义并研究 Euclid 空间中的弱 $1$-凸集与弱 $1$-半凸集，证明其 $1$-非凸性点集非空时为开集，且闭集的非空内部保持相应性质。

Topological properties of closed weakly $m$-convex sets

The present work considers the properties of generally convex sets in the $n$-dimensional real Euclidean space $\mathbb{R}^n$, $n>1$, known as weakly $m$-convex, $m=1,2,\ldots,n-1$. An open set of $\mathbb{R}^n$ is called weakly $m$-convex if for any boundary point of the set there exists an $m$-dimensional plane passing through this point and not intersecting the given set. A closed set of $\mathbb{R}^n$ is called weakly $m$-convex if it is approximated from the outside by a family of open weakly $m$-convex sets. A point of the complement of a set of $\mathbb{R}^n$ to the whole space is calle…

Recognizing the topology of the space of closed convex subsets of a Banach space

Let $X$ be a Banach space and $Conv_H(X)$ be the space of non-empty closed convex subsets of $X$, endowed with the Hausdorff metric $d_H$. We prove that each connected component of the space $Conv_H(X)$ is homeomorphic to one of the spaces: a singleton, the real line, a closed half-plane, the Hilbert cube multiplied by the half-line, the separable Hilbert space, or a Hilbert space of density not less than continuum.