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A partition of the unit sphere

• ✁

into regions of equal measure and small diameter

Paul Leopardi

paul.leopardi@unsw.edu.au

School of Mathematics, University of New South Wales. For presentation at Vanderbilt University, Nashville, November 2004.

A partition of the unit sphere

into regions of equal measure and small diameter – p. 1/26

The sphere

• Definition 1. For dimension
• , the unit sphere
• ✁

embedded in

is defined as

• ✁✝✆

Definition 2. Spherical polar coordinates describe a point

• f
• ✁

using one longitude,

, and

• ✛

colatitudes,

, for

• ✥

.

A partition of the unit sphere

into regions of equal measure and small diameter – p. 3/26

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

• f

measurable subsets of

• , such that for each

with

,

• ☎✠✡

and

A partition of the unit sphere

into regions of equal measure and small diameter – p. 4/26

Diameter bounded sets of partitions

Definition 4. The diameter of a region

is defined by

where

is the

Euclidean distance

. Definition 5. A set

• f partitions of
• ✠

is said to have diameter bound

if for all

, for each

, for

,

is said to be diameter bounded if there exists

such that

has diameter bound .

A partition of the unit sphere

into regions of equal measure and small diameter – p. 5/26

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

• ✏

,

• ☎

is diameter-bounded in the sense

• f Definition 5.

A partition of the unit sphere

into regions of equal measure and small diameter – p. 6/26

Precedents

The RZ partition is based on Zhou’s (1995) construction for

• ✎

as 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.

A partition of the unit sphere

into regions of equal measure and small diameter – p. 7/26

Stolarsky’s “Conjecture”

Stolasky (1973) asserts the existence of a diameter-bounded set of equal-measure partitions of

• ✁

for all

• , but offers no construction
• r 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.

A partition of the unit sphere

into regions of equal measure and small diameter – p. 8/26

Spherical zones, caps and collars

For

• ✂

, a zone can be described by

• ✁

where

.

is a North polar cap and

is a South polar cap. If

,

is a collar. For

• ✂

, the measure of a spherical cap of spherical radius

is

• ✎

where

• ✁

.

A partition of the unit sphere

into regions of equal measure and small diameter – p. 9/26

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

• ✛

;

.

A partition of the unit sphere

into regions of equal measure and small diameter – p. 10/26

RZ(3,99) Steps 1 to 2 θc ∆I V(θc) = VR = σ(S3)/99 ∆I = VR

1/3

RZ(3,99) Steps 3 to 5 θF,1 θF,2 ∆F y1 = 14.8... θF,3 ∆F y2 = 33.7... θF,4 ∆F y3 = 33.7... θF,5 ∆F y4 = 14.8... RZ(3,99) Steps 6 to 7 θ1 θ2 ∆1 m1 = 15 θ3 ∆2 m2 = 34 θ4 ∆3 m3 = 33 θ5 ∆4 m4 = 15 RZ(2,15) RZ(2,34) RZ(2,33) RZ(2,15)

Rounding the number of regions per collar

Similarly to Zhou (1995), given the sequence

for

• collars, with

define the sequences

and

by:

, and for

• ✥

,

Then

is the required number of regions in collar

, and we can show that

and

.

A partition of the unit sphere

into regions of equal measure and small diameter – p. 12/26

Geometry of regions

Each region

in collar

• f

• ✕

is of the form

• ✒

in spherical polar coordinates, where

• ✏

• ✒✂✁

, with

• ✁

. We can show that

where

and

.

A partition of the unit sphere

into regions of equal measure and small diameter – p. 13/26

The inductive step

Assuming that

• ✛

has diameter bound

• , define

• ✂

Then we can show that

A partition of the unit sphere

into regions of equal measure and small diameter – p. 14/26

Continuous analogs

Define

• ✆

,

• ✟

• ✟

• ✄✟✞

A partition of the unit sphere

into regions of equal measure and small diameter – p. 15/26

RZ(3,99) Steps 1 to 2 θc ∆I V(θc) = VR = σ(S3)/99 ∆I = VR

1/3

RZ(3,99) Steps 3 to 5 θF,1 θF,2 ∆F y1 = 14.8... θF,3 ∆F y2 = 33.7... θF,4 ∆F y3 = 33.7... θF,5 ∆F y4 = 14.8... RZ(3,99) Steps 6 to 7 θ1 θ2 ∆1 m1 = 15 θ3 ∆2 m2 = 34 θ4 ∆3 m3 = 33 θ5 ∆4 m4 = 15 RZ(2,15) RZ(2,34) RZ(2,33) RZ(2,15)

Properties of continuous analogs

For each collar

• ✥

, if we define

, then we can show that

A partition of the unit sphere

into regions of equal measure and small diameter – p. 17/26

Feasible domains

Define the feasible domain

• ✆

, where

• ✁

• ☎

• ☎

• ✛

• ✛

Assuming that

• ✛

has diameter bound

• , then for
• ✗

, for

in collar

• f

• ✕

, we can show

A partition of the unit sphere

into regions of equal measure and small diameter – p. 18/26

Properties and estimates of

• ✞

is smooth on

and is monotonic increasing in

.

• ✞

is positive and monotonic increasing in

.

• ✞

.

• For

and

,

• For

,

, where

• ✗

and

• ✏

A partition of the unit sphere

into regions of equal measure and small diameter – p. 19/26

Cap, , bounds

We can use properties and estimates of

to show that:

• There is a constant
• ✓

such that for

• ✂

, the diameter of each polar cap of

• ✕

is bounded by

• ✑

.

• For

• ✏

, if

• ✛

is diameter bounded, then there are constants

• ✓

• ✓

,

• ✕

• such

that for

• ✕

with

• ✝

• ✕

,

• ✑

A partition of the unit sphere

into regions of equal measure and small diameter – p. 20/26

Outline of proof of Theorem 2

Assume that

• ✗

and

• ✂

. Define

• ✆

• ✕

. Then if

• ✏

, if

• ✛

has diameter bound

• , and if
• , we have

• ✕

• ✑

, where

• ✆

• ✕

• ✁

. The diameter of any region is bounded by 2. Therefore for

• ,

• ✕

, where

• ✏

consists of equal segments, so

has diameter bound

. The result follows by induction.

A partition of the unit sphere

into regions of equal measure and small diameter – p. 21/26

Numerical results - constants

• ✁

• ✁

• ✗

• ✆

• ✂

• ☎

• ✂

• ✄

• ✂

Zhou obtains

for his (1995) algorithm.

A partition of the unit sphere

into regions of equal measure and small diameter – p. 22/26

10 10

1

10

2

10

3

10

4

10

5

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

Max diameter of region * N1/2 Upper bound Lower bound

N=number of regions

Bounds on maximum diameter coefficient for RZ partition of S2, 2004−09−22

10 10

1

10

2

10

3

10

4

10

5

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

Max diameter of region * N1/3 Upper bound Lower bound

N=number of regions

Bounds on maximum diameter coefficient for RZ partition of S3, 2004−09−22

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.

A partition of the unit sphere

into regions of equal measure and small diameter – p. 25/26

Illustration of RZ partition of

A partition of the unit sphere

into regions of equal measure and small diameter – p. 26/26