On the circumradius of acute point sets
نویسنده
چکیده
A set of points in IR n is said to be in acute position, if the angle determined by any triple of points is at most equal to =2. This paper considers the problem of computing the maximum circumradius of any set of n + 1 points in IR n that is in acute and general position and that has diameter one. It is shown that for n = 3, this maximum circumradius is at most equal to one. For arbitrary n, a lower bound of 1 4 p 2n on the maximum circumradius is proved.
منابع مشابه
Inequalities for Lattice Constrained Planar Convex Sets
Every convex set in the plane gives rise to geometric functionals such as the area, perimeter, diameter, width, inradius and circumradius. In this paper, we prove new inequalities involving these geometric functionals for planar convex sets containing zero or one interior lattice point. We also conjecture two results concerning sets containing one interior lattice point. Finally, we summarize k...
متن کاملComplete Systems of Inequalities
In this paper we summarize the known results and the main tools concerning complete systems of inequalities for families of convex sets. We discuss also the possibility of using these systems to determine particular subfamilies of planar convex sets with specific geometric significance. We also analyze complete systems of inequalities for 3-rotationally symmetric planar convex sets concerning t...
متن کاملIterated Nearest Neighbors and Finding Minimal Polytopes
We introduce a new method for nding several types of optimal k-point sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of the O(k) nearest neighbors to each point. We argue that this is better in a number of ways than previous algorithms, which were based on high order Voronoi diagrams. Our technique allows us for the rst time to e ciently maintain minima...
متن کاملON INEQUALITY Rp < R OF THE PEDAL TRIANGLE
In this paper we give a simple proof of the pedal triangle inequality Rp < R , where R is the circumradius of a triangle and Rp is the circumradius of the pedal triangle of an interior point with respect to this triangle. We also establish a stronger result and a refinement of inequality Rp < R . Some related interesting conjectures checked by the computer are put forward. Mathematics subject c...
متن کاملSome points on generalized open sets
The paper is an attempt to represent a study of limit points, boundary points, exterior points, border, interior points and closure points in the common generalized topological space. This paper takes a look at the possibilities of an extended topological space and it also considers the new characterizations of dense set.
متن کامل