Descriptive Proximities. Properties and Interplay Between Classical Proximities and Overlap

Abstract

The theory of descriptive nearness is usually adopted when dealing with subsets that share some common properties, even when the subsets are not spatially close. Set description arises from the use of probe functions to define feature vectors that describe a set; nearness is given by proximities. A probe on a nonempty set X is an n-dimensional, real-valued function that maps each member of X to its description. We establish a connection between relations on an object space X and relations on the corresponding feature space. In this paper, the starting point is what is known as $\mathcal {P}_\Phi$ proximity (two sets are $\mathcal {P}_\Phi$-near or $\Phi$-descriptively near if and only if their $\Phi$-descriptions intersect). We extend, elucidate and explain the connection between overlap and strong proximity in a theoretical approach to a more visual form of proximity called descriptive proximity, which leads to a number of applications. Descriptive proximities are considered on two different levels: weaker or stronger than the $\mathcal {P}_\Phi$ proximity. We analyze the properties and interplay between descriptions on the one hand and classical proximities and overlap relations on the other hand. Axioms and results for a descriptive Lodato strong proximity relation are given. A common descriptive proximity is an Efremovič proximity, whose underlying topology is $R_0$ (symmetry axiom) and Alexandroff-Hopf. For every description $\Phi$, any Čech, Lodato or EF $\Phi$-descriptive proximity is at the same time a Čech, Lodato or EF-proximity, respectively. But, the converse fails. A detailed practical application is given in terms of the construction of Efremovič descriptive proximity planograms, which complements recent operations research work on the allocation of shelf space in visual merchandising. Specific instances of applications of descriptive proximity are also cited.