Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1509.00447

Characterizations of compact sets in fuzzy sets spaces with $L_p$ metric

Huan Huang·Congxin Wu
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we present characterizations of totally bounded sets, relatively compact sets and compact sets in the fuzzy sets spaces $F_B(\mathbb{R}^m)$ and $F_B(\mathbb{R}^m)^p$ equipped with $L_p$ metric, where $F_B(\mathbb{R}^m)$ and $F_B(\mathbb{R}^m)^p$ are two kinds of general fuzzy sets on $\mathbb{R}^m$ which do not have any assumptions of convexity or star-shapedness. Subsets of $F_B(\mathbb{R}^m)^p$ include common fuzzy sets such as fuzzy numbers, fuzzy star-shaped numbers with respect to the origin, fuzzy star-shaped numbers, and the general fuzzy star-shaped numbers introduced by Qiu et al. The existed compactness criteria are stated for three kinds of fuzzy sets spaces endowed with $L_p$ metric whose universe sets are the former three kinds of common fuzzy sets respectively. Constructing completions of fuzzy sets spaces with respect to $L_p$ metric is a problem which is closely dependent on characterizing totally bounded sets. Based on preceding characterizations of totally boundedness and relatively compactness and some discussions on convexity and star-shapedness of fuzzy sets, we show that the completions of fuzzy sets spaces mentioned in this paper can be obtained by using the $L_p$-extension. We also clarify relation among all the ten fuzzy sets spaces discussed in this paper, which consist of five pairs of original spaces and the corresponding completions. Then, we show that the subspaces of $F_B(\mathbb{R}^m)$ and $F_B(\mathbb{R}^m)^p$ mentioned in this paper have parallel characterizations of totally bounded sets, relatively compact sets and compact sets. At last, as applications of our results, we discuss properties of $L_p$ metric on fuzzy sets space and relook compactness criteria proposed in previous work.

Explore related subjects

math.GM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Huan Huang, Congxin Wu. 2016-10-22. Characterizations of compact sets in fuzzy sets spaces with $L_p$ metric. https://arxiv.org/abs/1509.00447

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗
arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗