A New gH-Difference for Multi-Dimensional Convex Sets and Convex Fuzzy Sets

Luciano Stefanini,Barnabas Bede

doi:10.3390/axioms8020048

Abstract

In the setting of Minkowski set-valued operations, we study generalizations of the difference for (multidimensional) compact convex sets and for fuzzy sets on metric vector spaces, extending the Hukuhara difference. The proposed difference always exists and allows defining Pompeiu-Hausdorff distance for the space of compact convex sets in terms of a pseudo-norm, i.e., the magnitude of the difference set. A computational procedure for two dimensional sets is outlined and some examples of the new difference are given.

Highlights

It is well-known that in interval and set-valued arithmetic, the standard addition A + B ={ a + b| a ∈ A, b ∈ B} is not an invertible operation and in particular the algebraic difference A − B ={ a − b| a ∈ A, b ∈ B} is such that A − A 6= 0
The same problem applies to the general case of compact convex sets in Rn : finding a difference operation as an inverse of Minkowski addition A + B of compact convex sets has been a field of long interest; well-known and largely used examples are: the Hukuhara difference, proposed in [5], but it exists only in specific cases; the geometric Pontryagin difference, proposed in [6], but it may be the empty set; the Demyanov difference, introduced in the setting of subdifferential calculus and nonsmooth analysis
Extending the results in [12,14], we define a generalized difference for general compact convex sets and we extend it to fuzzy sets with compact and convex α-cuts

Summary

Introduction

Inversion of addition is important in set-valued and fuzzy arithmetic and analysis, with many applications e.g., in solving equations and differential equations (for recent results and other references to the fuzzy case, see e.g., [13,14,24,25,26,27,28,29,30,31,32,33,34]). Axioms 2019, 8, 48 has been addressed only occasionally and by very few papers in the literature; some basic results on the gH-difference for multidimensional intervals (boxes) were obtained in [12] and recently used by [35] in the study of fuzzy vector-valued functions.

The Space of Compact Convex Sets

A General Difference of Compact Convex Sets

Computation of the New Difference

Extension to Convex Fuzzy Sets

Conclusions