Determining the interset distance

Abstract

The following problem is considered. We are given a vector space that can be the vector space $\mathbb {R}^m$ or the vector space of symmetric $m \times m$ matrices $\mathbb {S}^m.$ There are two sets of vectors $\{a_i, 1 \le i \le r\}$ and $\{b_j, 1 \le j \le q\}$ in that vector space. Let K be some convex cone in the corresponding space. Let $a_i \ge _K b_j, \forall i,j,$ where $a_i \ge _K b_j$ mean that $a_i-b_j \in K.$ Let $\mathcal {A}_{\le }=\{x | a_i \ge _K x, \forall i, 1 \le i \le r \},$ where $a_i \ge _K x$ mean that $a_i-x \in K.$ Further let $\mathcal {B}_{\ge }=\{y | y \ge _K b_j, \forall j, 1 \le j \le q \}.$ In this work we study the question of finding and upperbounding the distance from the set $\mathcal {A}_{\le }$ to the set $\mathcal {B}_{\ge }$ in the case of cones $\mathbb {R}_+^m, \mathbb {L}^m, \mathbb {S}_+^m$.