From: Achill Schuermann Subject: Re: Caps on the sphere Date: Fri, 13 Oct 2000 12:51:46 +0000 Newsgroups: sci.math.research Summary: [missing] In article <39DDDFE0.22330517@po.cwru.edu>, Yossi Lonke wrote: >Suppose you have to cover as much surface area as possible of >the unit sphere in R^n, using spherical caps of the same radius >(i.e. intersections of Euclidean balls with the unit sphere, whose >centers are points of the sphere, all having the same radius), and let's >assume you have 2n points to center your caps at. The caps are allowed >to intersect, but all must be strictly smaller than half-a-sphere >(corresponding to radius = Sqrt(2)). My conjecture is that choosing >n points as an orthonormal basis and letting the other n points to be >their antipodes, yields a configuration giving maximal area. I just recently read (cf. [1], p.99, Remark 6.2.) about a remarkable result of Davenport and Hajos [2] about packings of spherical caps on the unit sphere in R^n, which might be interesting for You to look at: Let m(n,alpha) denote the maximal number of caps of "geodesic radius" alpha that can be packed on the unit sphere in R^n. Then | n+1, if pi/4 < alpha < pi/4+1/2*arcsin(1/n) m(n,alpha) = | | 2n, if alpha = pi/4 Note that Your conjecture is certainly true for alpha<=pi/4 and alpha>=arcos(1/sqrt(n)), as noticed by Robert Israel [although sqrt(2)/2 should be replaced by sqrt(2-sqrt(2)), I think]. Now the result above implies that for alpha larger than pi/4 there exist at least n-1 "overlappings". Maybe there is more information in reference [2]... I hope this helps - Achill Schuermann. [1] Chuanming Zong, Sphere Packings, Springer, New York, 1999. [2] Davenport, H. and Hajos, G., Problem 35, Mat. Lapok 2 (1951), 68. \|/ 0^0 --------------------oOo--(_)--oOo------------- Achill Schuermann University of Siegen Germany