On Some Idempotent and Non-Associative Convex Structure

TL;DR

This paper extends the concept of $ ext{B}$-convexity from subsets of $ ext{R}^n_+$ to the entire Euclidean space by defining a new algebraic structure and demonstrating the existence of convex hull limits.

Contribution

It introduces a unital idempotent non-associative magma and an extended $n$-ary operation to generalize $ ext{B}$-convexity across all of $ ext{R}^n$.

Findings

Existence of Kuratowski-Painlevé limit of convex hulls in $ ext{R}^n$

Explicit extension of $ ext{B}$-convexity to the whole Euclidean space

Development of a new algebraic structure for convexity analysis

Abstract

$\mathbb B$-convexity was defined in [7] as a suitable Kuratowski-Painlev\'e upper limit of linear convexities over a finite dimensional Euclidean vector space. Excepted in the special case where convex sets are subsets of $\mathbb R^n_ +$, $\mathbb B$-convexity was not defined with respect to a given explicit algebraic structure. This is done in that paper, which proposes an extension of $\mathbb B$-convexity to the whole Euclidean vector space. An unital idempotent and non-associative magma is defined over the real set and an extended $n$-ary operation is introduced. Along this line, the existence of the Kuratowski-Painlev\'e limit of the convex hull of two points over $\mathbb R^n$ is shown and an explicit extension of $\mathbb B$-convexity is proposed.