Related papers: Locally nonconical convexity

A convex subset X of a linear topological space is called compactly convex if there is a continuous compact-valued map $\Phi:X\to exp(X)$ such that $[x,y]\subset\Phi(x)\cup \Phi(y)$ for all $x,y\in X$. We prove that each convex subset of…

A topological space is locally equiconnected if there exists a neighborhood $U$ of the diagonal in $X\times X$ and a continuous map $\lambda:U\times[0,1]\to X$ such that $\lambda(x,y,0)=x$, $\lambda(x,y,1)=y$ et $\lambda(x,x,t)=x$ for…

Motivated by ill-posed PDEs such as $\mathrm{div} (v) = F$ we study locally convex topologies $\mathcal{T}_{\mathcal{C}}$ on real vector spaces $X$ that are a ``localized'' version of a locally convex topology $\mathcal{T}$ to members of a…

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…

We study the geometric structure of compact convex sets in 2-dimensional asymmetric normed lattices. We prove that every q-compact convex set is strongly q-compact and we give a complete geometric description of the compact convex sets with…

Necessary and sufficient conditions for convexity and strong convexity, respectively, of sublevel sets that are defined by finitely many real-valued $C^{1,1}$-maps are presented. A novel characterization of strongly convex sets in terms of…

This paper presents new approaches to the fixed point property for nonexpansive mappings in L^1 spaces. While it is well-known that L^1 fails the fixed point property in general, we provide a complete and self-contained proof that…

We construct a complete locally convex topological vector space $X$ of countable algebraic dimension and a continuous linear operator $T:X\to X$ such that $T$ has no non-trivial closed invariant subspaces.

Fix any $\lambda\in\mathbb{C}$. We say that a set $S\subseteq\mathbb{C}$ is $\lambda$-$convex$ if, whenever $a$ and $b$ are in $S$, the point $(1-\lambda)a+\lambda b$ is also in $S$. If $S$ is also (topologically) closed, then we say that…

In many applications it is important to establish if a given topological preordered space has a topology and a preorder which can be recovered from the set of continuous isotone functions. Under antisymmetry this property, also known as…

We give a proof to the following theorem, which is well-known among experts: A connected subcomplex $W$ of a finite dimensional CAT(0) cubed complex $X$ is convex if and only if Lk$(v, W)$ is a full subcomplex of Lk$(v, X)$ for every vertex…

This paper investigates the convexity of the solution set of the linear complementarity problems over tensor spaces (TLCPs). We introduce the notion of a $T$-column sufficient tensor and study its properties and relationships with several…

The classical theory of regularity of embeddings of compact convex sets was developed in the 1970s, exclusively in the real case, and even there it does not appear to have been stated in its simplest form. We begin by revisiting this…

For the study of some typical problems in finance and economics, \v{Z}itkovi\'{c} %[G. \v{Z}itkovi\'{c}, Convex compactness and its applications, Math. Finan. Eco., 3(1)(2010) 1--12] introduced convex compactness and gave many remarkable…

Properties of several sorts of lattices of convex subsets of R^n are examined. The lattice of convex sets containing the origin turns out, for n>1, to satisfy a set of identities strictly between those of the lattice of all convex subsets…

While there is a well developed theory of locally solid topologies, many important convergences in vector lattice theory are not topological. Yet they share many properties with locally solid topologies. Building upon the theory of…

We study the smallest convex lattice generated by a finite set of points. To analyze this structure, we introduce the notion of a point configuration, defined via the relative lattice. Under a suitable completeness condition, this lattice…

We study when a piecewise full group (a.k.a. topological full group) of homeomorphisms of the Cantor space $X$ can be given a non-discrete totally disconnected locally compact (t.d.l.c.) topology and give a criterion for the alternating…

In this note we show that a connected, closed and locally convex subset (with an extra assumption on the diameter with respect to the induced length metric if $\kappa>0$) of a $CAT(\kappa)$ space is convex.

This article explores the model-dependent nature of set cardinality, emphasizing that cardinality is not absolute but varies across different axiomatic frameworks. Although Cantor's diagonal argument shows the real numbers are…