into regions of equal measure and small diameter Paul Leopardi paulleopardi@unsweduau School of Mathematics, University of New South Wales For presentation at Vanderbilt University, Nashville, November 2004 into regions of equal measure and small diameter p 1/2
The sphere Definition 1 For dimension is defined as, the unit sphere embedded in Definition 2 Spherical polar coordinates describe a point using one longitude,, and colatitudes,, for of into regions of equal measure and small diameter p 3/2
Equal-measure partitions Definition 3 Let be a measurable set and a measure with An equal-measure partition of for is a nonempty finite set measurable subsets of, such that for each with, of and into regions of equal measure and small diameter p 4/2
Diameter bounded sets of partitions Definition 4 The diameter of a region is defined by where is the Euclidean distance Definition 5 A set diameter bound, of partitions of if for all is said to have, for each, for is said to be diameter bounded if there exists has diameter bound such that into regions of equal measure and small diameter p 5/2
Key properties of the RZ partition of The recursive zonal (RZ) partition of into regions is denoted as The set of partitions The RZ partition satisfies the following theorems Theorem 1 For dimension, let be the usual surface measure on inherited from the Lebesgue measure on via the usual embedding of in Then for, is an equal-measure partition for Theorem 2 For of Definition 5, is diameter-bounded in the sense into regions of equal measure and small diameter p 6/2
Precedents The RZ partition is based on Zhou s (1995) construction for modified by Ed Saff, and on Ian Sloan s sketch of a partition of (2003) Alexander (1972) uses the existence of a diameter-bounded set of equal-area partitions of to analyse the maximum sum of distances between points Alexander (1972) suggests a construction different from Zhou (1995) Equal-area partitions of used in the geosciences and astronomy do not have a proven bound on the diameter of regions as into regions of equal measure and small diameter p 7/2
Stolarsky s Conjecture Stolasky (1973) asserts the existence of a diameter-bounded set of equal-measure partitions of for all, but offers no construction or existence proof Beck and Chen (1987) quotes Stolarsky Bourgain and Lindenstrauss (1988) quotes Beck and Chen Wagner (1993) implies the existence of an RZ-like construction for Bourgain and Lindenstrauss (1993) gives a partial construction into regions of equal measure and small diameter p 8/2
Spherical zones, caps and collars For, a zone can be described by where If is a North polar cap and, is a collar is a South polar cap For, the measure of a spherical cap of spherical radius is where into regions of equal measure and small diameter p 9/2
Outline of the RZ algorithm The RZ algorithm is recursive in dimension Algorithm for : There is a single region which is the whole sphere; Divide the circle into equal segments; Divide the sphere into zones, each the same measure as an integer number of regions: North and South polar spherical caps and a number of spherical collars; Partition each spherical collar into regions of equal measure, using the RZ algorithm for dimension ; into regions of equal measure and small diameter p 10/2
RZ(3,99) Steps 1 to 2 RZ(3,99) Steps 3 to 5 θ F,1 θ c I y 1 = 148 θ F,2 V(θ c ) = V R = σ(s 3 )/99 I = V R 1/3 y 2 = 337 y 3 = 337 F F F F θ F,3 y 4 = 148 θ F,4 θ F,5 RZ(3,99) Steps 6 to 7 θ 1 RZ(2,15) RZ(2,34) m 1 = 15 θ 2 m 2 = 34 1 2 θ 3 RZ(2,33) RZ(2,15) m 3 = 33 3 4 m 4 = 15 θ 4 θ 5
Rounding the number of regions per collar Similarly to Zhou (1995), given the sequence for collars, with define the sequences, and for and by:, Then show that is the required number of regions in collar and, and we can into regions of equal measure and small diameter p 12/2
Geometry of regions is of the form of in collar Each region in spherical polar coordinates, where, with We can show that and where into regions of equal measure and small diameter p 13/2
The inductive step has diameter bound, define Assuming that Then we can show that into regions of equal measure and small diameter p 14/2
Continuous analogs, Define into regions of equal measure and small diameter p 15/2
RZ(3,99) Steps 1 to 2 RZ(3,99) Steps 3 to 5 θ F,1 θ c I y 1 = 148 θ F,2 V(θ c ) = V R = σ(s 3 )/99 I = V R 1/3 y 2 = 337 y 3 = 337 F F F F θ F,3 y 4 = 148 θ F,4 θ F,5 RZ(3,99) Steps 6 to 7 θ 1 RZ(2,15) RZ(2,34) m 1 = 15 θ 2 m 2 = 34 1 2 θ 3 RZ(2,33) RZ(2,15) m 3 = 33 3 4 m 4 = 15 θ 4 θ 5
Properties of continuous analogs, if we define, then we can show that For each collar into regions of equal measure and small diameter p 17/2
Feasible domains, where Define the feasible domain has diameter bound, then for of, we can show in collar Assuming that, for into regions of equal measure and small diameter p 18/2
Properties and estimates of is smooth on and is monotonic increasing in is positive and monotonic increasing in For and, For,, where and into regions of equal measure and small diameter p 19/2
Cap,, bounds We can use properties and estimates of There is a constant such that for diameter of each polar cap of For, if there are constants that for with to show that:, the is bounded by is diameter bounded, then, such, into regions of equal measure and small diameter p 20/2
Outline of proof of Theorem 2 Assume that Define and Then if, if, we have has diameter bound, and if, where The diameter of any region is bounded by 2 Therefore for,, where consists of equal segments, so diameter bound The result follows by induction has into regions of equal measure and small diameter p 21/2
Numerical results - constants Zhou obtains for his (1995) algorithm into regions of equal measure and small diameter p 22/2
Bounds on maximum diameter coefficient for RZ partition of S 2, 2004 09 22 8 75 7 65 6 55 5 Upper bound 45 Lower bound 4 35 Max diameter of region * N 1/2 3 25 2 10 0 10 1 10 2 10 3 10 4 10 5 N=number of regions
Bounds on maximum diameter coefficient for RZ partition of S 3, 2004 09 22 8 75 7 65 6 55 5 Upper bound 45 Lower bound 4 35 Max diameter of region * N 1/3 3 25 2 10 0 10 1 10 2 10 3 10 4 10 5 N=number of regions
Stereographic projection of to In Cartesian coordinates, the stereographic projection is if When restricted to, The north pole projects to The south polar cap projects to a ball Collars project to differences between balls Spheres project to generalized spheres into regions of equal measure and small diameter p 25/2
Illustration of RZ partition of into regions of equal measure and small diameter p 26/2