All-maximum and all-minimum problems under some measures

In this paper we investigate the following type of proximity problems: given a set of n   points in the plane P={p1,p2,p3,…,pn}P={p1,p2,p3,…,pn}, for each point pipi find a pair {pj,pk}{pj,pk}, where i≠j≠ki≠j≠k, such that a measure MM defined on the triplet of points {pi,pj,pk}{pi,pj,pk} is maximized or minimized. The cases where MM is the distance from pipi to the segment or line defined by {pj,pk}{pj,pk} have been extensively studied. We study the cases where MM is the sum, product or the difference of the distances from pipi to the points pjpj and pkpk; distance from pipi to the line defined by pjpj and pkpk; the area, perimeter of the triangle defined by pipi, pjpj and pkpk, as well as the radius of the circumcircle defined by them. We also discuss the all-farthest triangle problem in the triangle-distance measure when P is a set of points in 3 dimensions.