J. Geom. 103 (2012), 125–129c© 2012 Springer Basel AG0047-2468/12/010125-5published online April 17, 2012DOI 10.1007/s00022-012-0113-7 Journal of Geometry

Extremal problems for the central projection

Shigehiro Sakata

Abstract. We consider area minimizing problems for the image of a closedsubset in the unit sphere under a projection from the center of the sphereto a tangent plane, the central projection. We show, for any closed subsetin the sphere, the uniqueness of a tangent plane that minimizes the area,and then the minimality of the spherical discs among closed subsets withthe same spherical area.

Mathematics Subject Classification. 51M16, 51M25.

Keywords. Central projection, polar set, gnomonic projection, extremalproblem.

1. Introduction

Let Sn denote the n-dimensional unit sphere, Sn = {x ∈ Rn+1||x| = 1}, σn

the spherical standard measure on Sn, and · the standard inner product ofR

n+1. Let Ω be a closed subset in Sn obtained as the closure of an open setand Ω∗ = ∩y∈Ω{x ∈ Sn|x · y ≤ 0}, which is called the polar set of Ω. In thefollowing, we always assume that the interior of Ω∗ is not empty, namely, Ω is

contained in the interior of a hemisphere. Let px (x ∈ −◦

Ω∗, where◦

Ω∗ denotesthe interior of Ω∗) be the projection from Ω to the tangent plane of Sn at x,

and AΩ the map that assigns the area of px(Ω) to a point x in −◦

Ω∗:

px : Ω � y �→ y

x · y∈ TxSn, AΩ(x) = Area(px(Ω)).

This projection px is used for making a (local) world atlas and also called thegnomonic projection when n = 2 in geography.

Since the Jacobian of px(y) is given by Jpx(y) = 1(x·y)n+1 , we have

AΩ(x) =∫

Ω

1(x · y)n+1

dσn(y).

126 S. Sakata J. Geom.

Rn

{h}

Rn

{0}

Tx S n

0

x

A (x)

Ω

∗ΩS n S n

Ω

Figure 1 The left and right figures express the centralprojection and the solid angle, respectively

Gao et al. showed the followings in [1]. First they showed the existence of apoint that gives the minimum value of AΩ when Ω is convex in Sn, but didnot discuss the uniqueness. Note that Ω is said to be convex in Sn if the cone∪y∈Ω{z ∈ R

n+1|z = ty ∃t ∈ [0, 1]} is convex in Rn+1. It follows from the fact

that AΩ(x) is continuous on −◦

Ω∗ and diverges to +∞ as x approaches theboundary of −Ω∗. Second they gave the characterization of such a point. Bydirect calculation, they showed that if x gives the minimum value of AΩ, thenx satisfies the following formula:

x =1

AΩ(x)

∫

Ω

y

(x · y)n+2dσn(y) =

1Area(px(Ω))

∫

px(Ω)

zdz.

Hence such a point x coincides with the centroid of px(Ω) in TxSn. Moreover,they showed the minimality of the spherical discs among spherical convexbodies with the same spherical area.

In [3] the author considered the solid angle of a closed subset in Rn at a point

in the level set of height h > 0 and studied the uniqueness of a maximum pointof the solid angle, which is a function of the observation point in R

n × {h}. Inthe solid angle case, the uniqueness of such a point does not always hold. Infact, if the closed subset is a disjoint union of two discs of the same size, andthe height h is small enough, then there exist (at least) two maximum pointsof the solid angle.

The uniqueness of a minimum point of AΩ can be regarded as an inverse prob-lem for that of the solid angle (see Fig. 1). In this paper we show the uniquenessof an AΩ minimizer without the assumption of Ω being convex. Furthermore,we show the minimality of the spherical discs among closed subsets with thesame spherical area. We also study similar problems for the hyperbolic space.

Vol. 103 (2012) Extremal problems for the central projection 127

2. Extremal points of AΩ(x)

Let Ω and AΩ be as in the introduction. Let Dε(y) denote the spherical disc{x ∈ Sn|distSn(x, y) ≤ ε}. We understand that Ω satisfies the the Poincarecondition inside if for any point y in the boundary of Ω, there exist positiveconstants ε and θ such that Dε(y) ∩ Ω contains a spherical circular cone withvertex y and angle θ.

Theorem 2.1. (1) There exists a minimum point of AΩ in −Ω∗. In particu-lar, if Ω satisfies the Poincare condition inside, then the minimum pointis contained in the interior of −Ω∗.

(2) If Ω satisfies the Poincare condition inside, then AΩ has a unique mini-mum point.

Proof. (1) The existence follows from the fact that the function AΩ is continu-ous on the compact set −Ω∗ in the extended sense. In particular, if Ω satisfiesthe Poincare condition inside, then AΩ diverges to +∞ as x approaches theboundary of −Ω∗.

(2) We show the uniqueness of an AΩ minimizer. Remark that −Ω∗ =∩y∈ΩDπ

2(y) is convex in Sn since an intersection of convex sets in Sn is convex

in Sn. By using polar coordinates, we can put

x = x(θ1, . . . , θn)= (cos θ1, sin θ1 cos θ2, . . . , sin θ1 · · · sin θn−1 cos θn, sin θ1 · · · sin θn−1 sin θn),

where (θ1, . . . , θn−1, θn) ∈ [0, π]×· · ·× [0, π]× [0, 2π]. Since ∂2x∂θ2

1= −x, we have

∂2AΩ

∂θ21

(x) = (n + 1)∫

Ω

(x · y)2 + (n + 2)( ∂x∂θ1

· y)2

(x · y)n+3dσn(y) > 0

for any x ∈ −◦

Ω∗. Suppose that there exist two points x′ and x′′ that give theminimum value of AΩ. By a rotation of Sn, we may assume that the sub-arcbetween x′ and x′′ of the great circle has constant θi coordinates for i �= 1.Then we have

∂AΩ

∂θ1(x′) =

∂AΩ

∂θ1(x′′) = 0,

which is a contradiction since the arc is contained in −Ω∗. �Remark 2.2. We assumed that Ω satisfies the Poincare condition inside for theuniqueness of the AΩ minimizer. We can replace the assumption by “AΩ doesnot have minimum points in the boundary of −Ω∗” for obtaining the sameresult as in the second assertion of Theorem 2.1.

Corollary 2.3. Suppose that Ω satisfies the Poincare condition inside. If Ω ispoint symmetric about a point xs, then xs is the unique minimum point of AΩ.

Example. (1) Let Ω be a (disjoint) union of two spherical discs. By Theo-rem 2.1, AΩ has a unique minimum point.

128 S. Sakata J. Geom.

(2) By Corollary 2.3, if the two discs in (1) above have the same size, thenthe middle point of the sub-arc between two centers of discs is the uniqueAΩ minimizer.

(3) By Corollary 2.3, the center of a disc D in Sn is the unique minimumpoint of AD. This fact can also be indicated by the moving plane method([2]) because of the symmetry in the integrand of the partial derivativeof AD.

Theorem 2.4. Let D be a disc in Sn. If σn(Ω) = σn(D), then

minx∈−

◦D∗

AD(x) ≤ minx∈−

◦Ω∗

AΩ(x)

and equality holds if and only if Ω is a disc in Sn.

Proof. By a translation of D, we may assume that the center of D coincideswith a minimum point of AΩ. Let xc denote the center of D. Then we obtain

AΩ(xc) − AD(xc)

=∫

Ω\(Ω∩D)

1(xc · y)n+1

dσn(y) −∫

D\(Ω∩D)

1(xc · y)n+1

dσn(y)

≥ 0

since the following inequality holds for y′ ∈ Ω\(Ω ∩ D) and y′′ ∈ D\(Ω ∩ D):

xc · y′ = cos ∠(xc, y′) ≤ cos ∠(xc, y

′′) = xc · y′.

Equality holds if and only if σn(Ω\(Ω∩D)) = 0, namely, Ω is a disc in Sn. �

3. Hyperbolic case

Let 〈·, ·〉 denote the indefinite inner product of Rn+1 given by 〈x, y〉 = x1y1 +

· · · + xnyn − xn+1yn+1, Rn+11 the (n + 1)-dimensional Euclidean space with

〈·, ·〉, and Hn the Lorentz model of the n-dimensional hyperbolic space, H

n ={x ∈ R

n+11 |〈x, x〉 = −1, xn+1 > 0}. Let Ω be a compact subset in H

n obtainedas the closure of an open set. We define the maps px (x ∈ H

n) and AΩ in thesame way as in the spherical case:

px : Ω � y �→ − y

〈x, y〉 ∈ TxHn, AΩ(x) = Area(px(Ω)).

Since the Jacobian of px(y) is given by Jpx(y) = 1(−〈x,y〉)n+1 , we have

AΩ(x) =∫

Ω

1(−〈x, y〉)n+1

dμn(y),

where μn is the standard hyperbolic measure on Hn.

The existence of an AΩ maximizer follows from the fact that AΩ is continuouson H

n and converges to 0 as xn+1 goes to +∞. The same computation as in [1]

Vol. 103 (2012) Extremal problems for the central projection 129

shows that if x gives the maximum value of AΩ, then x satisfies the followingformula:

x =1

AΩ(x)

∫

Ω

y

(−〈x, y〉)n+2dμn(y) =

1Area(px(Ω))

∫

px(Ω)

zdz.

The uniqueness of such a point does not always hold. For example, the follow-ing set Ω in H

1 has two maximum points of AΩ:

Ω = {(sinh θ, cosh θ)| − 2 ≤ θ ≤ −1} ∪ {(sinh θ, cosh θ)|1 ≤ θ ≤ 2}.

On the other hand, we can obtain the following theorem corresponding toTheorem 2.4 with the same argument:

Theorem 3.1. Let D be a disc in Hn. If μn(Ω) = μn(D), then we have

maxx∈Hn

AΩ(x) ≤ maxx∈Hn

AD(x)

and equality holds if and only if Ω is a disc in Hn.

Acknowledgments

The author would like to express his deep gratitude to his advisors Jun O’Haraand Gil Solanes for giving kind advice to him. The author also thanks to thereferee for careful reading.

References

[1] Gao, F., Hug, D., Schneider, R.: Intrinsic volumes and polar sets in sphericalspaces. Math. Notae. 41, 159–176 (2001/2002)

[2] Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry and related properties via the max-imum principle. Comm. Math. Phys. 68, 209–243 (1979)

[3] Sakata, S.: Extremal problems for the solid angle. arXiv:1103.1727

Shigehiro SakataDepartment of Mathematics and Information ScienceTokyo Metropolitan University1-1 Minami Osawa, Hachiouji-ShiTokyo 192-0397Japane-mail: sakata-shigehiro@ed.tmu.ac.jp

Received: December 8, 2011.

Revised: March 13, 2012.