Isosceles Triple Convexity

A set S C (?) Rd is called it-convex if, for any two distinct points x, y∈S, there is z∈S, such that the convex hull of x, y, z forms an isosceles triangle (maybe a degenerated isosceles triangle), i.e., one of the three points is equidistant from the others.In the thesis we first give some non-discrete it-convex sets according to the definition. Then we probe into the it-convexity about discrete point sets in the plane, and obtain the fol-lowing results:(1) we discuss the it-convexity about the vertex sets of eleven Archimedean tilings (infinite sets), and prove that the vertex sets of Archimedean tillings (36),(44),(63),(3.6.3.6),(32.4.3.4),(34.6) are it-convex sets, but the vertex sets of Archimedean tillings (4.82),(3.4.6.4),(33.42),(4.6.12),(3.122) are not it-convex sets.(2) We study the it-convexity about finite subsets of the planar integral lattice, and prove that for any two distinct points xk,l=(k, l) xm,n=(m, n) in the integral lattice, and any two shortest lattice paths from xk,l to xm,n, there are8and only8it-convex sets among the family of the point sets including all lattice points lying on or inside the region bounded by the two paths.(3) We investigate the it-convexity about the general discrete point sets, and provide a condition to determine the it-convexity of the vertex set of any regular n-polygon. Furthermore, we obtain a lower bound for the number of isosceles triangles contained in the minimum n-point it-convex sets.