A characterization theorem and an algorithm for a convex hull problem

Given $$S= \{v_1, \dots , v_n\} \subset {\mathbb {R}}^m$$S={v1,⋯,vn}⊂Rm and $$p \in {\mathbb {R}}^m$$p∈Rm, testing if $$p \in conv(S)$$p∈conv(S), the convex hull of $$S$$S, is a fundamental problem in computational geometry and linear programming. First, we prove a Euclidean distance duality, distinct from classical separation theorems such as Farkas Lemma: $$p$$p lies in $$conv(S)$$conv(S) if and only if for each $$p' \in conv(S)$$p′∈conv(S) there exists a pivot, $$v_j \in S$$vj∈S satisfying $$d(p',v_j) \ge d(p,v_j)$$d(p′,vj)≥d(p,vj). Equivalently, $$p \not \in conv(S)$$p∉conv(S) if and only if there exists a witness, $$p' \in conv(S)$$p′∈conv(S) whose Voronoi cell relative to $$p$$p contains $$S$$S. A witness separates $$p$$p from $$conv(S)$$conv(S) and approximate $$d(p, conv(S))$$d(p,conv(S)) to within a factor of two. Next, we describe the Triangle Algorithm: given $$\epsilon \in (0,1)$$ϵ∈(0,1), an iterate, $$p' \in conv(S)$$p′∈conv(S), and $$v \in S$$v∈S, if $$d(p, p') < \epsilon d(p,v)$$d(p,p′)<ϵd(p,v), it stops. Otherwise, if there exists a pivot $$v_j$$vj, it replace $$v$$v with $$v_j$$vj and $$p'$$p′ with the projection of $$p$$p onto the line $$p'v_j$$p′vj. Repeating this process, the algorithm terminates in $$O(mn \min \{ \epsilon ^{-2}, c^{-1}\ln \epsilon ^{-1} \})$$O(mnmin{ϵ-2,c-1lnϵ-1}) arithmetic operations, where $$c$$c is the visibility factor, a constant satisfying $$c \ge \epsilon ^2$$c≥ϵ2 and $$\sin (\angle pp'v_j) \le 1/\sqrt{1+c}$$sin(∠pp′vj)≤1/1+c, over all iterates $$p'$$p′. In particular, the geometry of the input data may result in efficient complexities such as $$O(mn \root t \of {\epsilon ^{-2}} \ln \epsilon ^{-1})$$O(mnϵ-2tlnϵ-1), $$t$$t a natural number, or even $$O(mn \ln \epsilon ^{-1})$$O(mnlnϵ-1). Additionally, (i) we prove a strict distance duality and a related minimax theorem, resulting in more effective pivots; (ii) describe $$O(mn \ln \epsilon ^{-1})$$O(mnlnϵ-1)-time algorithms that may compute a witness or a good approximate solution; (iii) prove generalized distance duality and describe a corresponding generalized Triangle Algorithm; (iv) prove a sensitivity theorem to analyze the complexity of solving LP feasibility via the Triangle Algorithm. The Triangle Algorithm is practical and competitive with the simplex method, sparse greedy approximation and first-order methods. Finally, we discuss future work on applications and generalizations.