Determining the interset distance

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