Related papers: Nonlinear Cone Separation Theorems in Real Topolog…
In this paper, we derive some new results for the separation of two not necessarily convex cones by a (convex) cone / conical surface in real (reflexive) normed spaces. In essence, we follow the nonlinear and nonsymmetric separation…
In this paper, we study relationships between symmetric and non-symmetric separation of (not necessarily convex) cones by using separating cones of Bishop-Phelps type in real normed spaces. Besides extending some known results for the…
We present a concise proof for the supporting hyperplane theorem. We then observe that the proof not only establishes the supporting hyperplane theorem but also extends it to a hyperplane separation theorem for certain non-convex sets. The…
A closed convex subset of a normed linear space is said to have the strong separation property if it can be strongly separated from every other disjoint closed and convex set by a closed hyperplane. In this paper we give some results on the…
This paper deals with $\varepsilon$-efficient and $\varepsilon$-properly efficient points with respect to a co-radiant set in vector optimization problems. In the first part of the paper, we establish a new nonlinear separation theorem for…
In this paper, we deal with analytic and geometric properties of orthogonally convex sets. We establish a Blaschke-type theorem for path-connected and orthogonally convex sets in the plane using orthogonally convex paths. The separation of…
In this paper, we establish a suitable version of the Hahn-Banach theorem within the framework of Colombeau spaces, a class of spaces used to model generalized functions. Our approach addresses the case where maps are defined…
Concept of bi-soft topological spaces is introduced. Several notions of a soft topological space are generalized to study bi-soft topological spaces. Separation axioms play a vital role in study of topological spaces. These concepts have…
It is well-known that scalarization techniques (e.g., in the sense of Gerstewitz; Kasimbeyli; Pascoletti and Serafini; Zaffaroni) are useful for generating (weakly, properly) efficient solutions of vector optimization problems. One…
In this short note we provide a new proof of the recent result of Han and Nashimura on the separation of spherical convex sets established in arXiv:2002.06558. Our proof is based on a result stated in locally convex spaces.
We prove two theorems on the locally finite decompositions of the cones of divisors by the cones which correspond to canonical and minimal models. We introduce the concept of the numerical linear systems in order to simplify the argument on…
This study focuses on defining normal and strictly convex structures within Menger cone PM-space. It also presents a shared fixed point theorem for the existence of two self-mappings constructed on a strictly convex probabilistic cone…
The concept of separation by hyperplanes is fundamental for convex geometry and its tropical (max-plus) analogue. However, analogous separation results in max-min convex geometry are based on semispaces. This paper answers the question…
Two planar sets are circularly separable if there exists a circle enclosing one of the sets and whose open interior disk does not intersect the other set. This paper studies two problems related to circular separability. A linear-time…
In the study of soft topological spaces, two types of separation axioms have given if soft points and single point soft sets have been taken as separated objects respectively. In this paper, some examples and properties are given to explore…
The classic double bubble theorem says that the least-perimeter way to enclose and separate two prescribed volumes in $\mathbb{R}^N$ is the standard double bubble. We seek the optimal double bubble in $\mathbb{R}^N$ with density, which we…
In this paper, two parallel notions of convexity of sets are introduced in the abelian semigroup setting. The connection of these notions to algebraic and to set-theoretic operations is investigated. A formula for the computation of the…
Functions with uniform level sets can represent orders, preference relations or other binary relations and thus turn out to be a tool for scalarization that can be used, e.g., in multicriteria optimization, decision theory, mathematical…
A pair of non-adjacent edges is said to be separated in a circular ordering of vertices, if the endpoints of the two edges do not alternate in the ordering. The circular separation dimension of a graph $G$, denoted by $\pi^\circ(G)$, is the…
We observe that many of the separation axioms of topology (including $T_0-T_4$) can be expressed concisely and uniformly in terms of category theory as lifting properties (in the sense of Quillen model categories) with respect to (usually…