Descriptive Proximities. Properties and Interplay Between Classical Proximities and Overlap

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 (Formula presented.) proximity (two sets are (Formula presented.)-near or (Formula presented.)-descriptively near if and only if their (Formula presented.)-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 (Formula presented.) 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 EfremoviÄ\u8d proximity, whose underlying topology is (Formula presented.) (symmetry axiom) and Alexandroff-Hopf. For every description (Formula presented.), any Ä\u8cech, Lodato or EF (Formula presented.)-descriptive proximity is at the same time a Ä\u8cech, Lodato or EF-proximity, respectively. But, the converse fails. A detailed practical application is given in terms of the construction of EfremoviÄ\u8d 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