BLASCHKE S ROLLING BALL THEOREM AND THE TRUDINGER-WANG MONOTONE BENDING. 1. Introduction

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1 BLASCHKE S ROLLING BALL THEOREM AND THE TRUDINGER-WANG MONOTONE BENDING ARAM L. KARAKHANYAN arxiv:5.086v math.ap] 7 Dec 05 Abstract. We revisit the classical rolling ball theorem of Blaschke for convex surfaces with positive curvature and show that it is linked to another inclusion principle in the optimal mass transportation theory due to Trudinger and Wang. We also discuss an application to reflector antennae design problem.. Introduction In this note we give two applications of an inclusion principle known as the rolling ball Theorem of Blaschke. Let M and M be two hypersurfaces in R d. We say that M and M are internally tangent at x M if they are tangent at x and have the same outward normal. Denote by II x M the second fundamental form of M at x and let n(x) be the outward unit normal at x. Then we have Theorem.. Suppose M and M are smooth convex surfaces with strictly positive scalar curvature such that II x M II x M for all x M, x M such that n(x) = n (x ). If M and M are internally tangent at one point then M is contained in the convex region bounded by M. W. Blaschke ], pp. 4-7 proved Theorem. for closed curves in R. D. Koutroufiotis 5] generalized Blaschke s theorem for complete curves in R and complete surfaces in R 3. Later J. Rauch 9], by using Blaschke s techniques, proved this result for compact surfaces in R d and J.A. Delgado 3] for complete surfaces. Finally J. N. Brooks and J. B. Strantzen generalized Blaschke s theorem for non-smooth convex sets showing that the local inclusion implies global inclusion ]. Observe that if M and M are internally tangent at x, then a necessary condition for M to be inside M near x is (.) II x (v) II x(v) for all v T x M = T x M. The tangent planes are parallel because M and M are internally tangent at x. Therefore Theorem. says that if for all x M, x M, x x with coinciding normals n (x ) = n(x) such that after translating M by x x we have that the translated surface M is 000 Mathematics Subject Classification. Primary 5A0, 49Q0, 90B06. Keywords: Optimal transport, inclusion principle, Blaschke s rolling ball theorem.

2 ARAM L. KARAKHANYAN locally inside M then M is globally inside M. In other words, the local inclusion implies global inclusion or M rolls freely inside M. Our aim is to apply Theorem. to optimal transportation theory and reflector antennae design problems. More specifically, for a smooth cost function c : R d R d R (subject to some standard conditions including the weak A3 condition) and a pair of bounded smooth convex domains U, V R d such that U is c convex with respect to V (see Definition. below), we would like to take M = U to be the reference domain and M = N := {x R d s.t. c(x, y 0 ) = c(x, y ) + a}, y, y V for some constant a. Then M is the boundary of sub-level set of the cost function c. We prove that if U is locally inside N in above sense then U is globally inside N provided that the sets N are convex for all y, y, a. The precise result is formulated in Theorem 3. below and applications in Section 4. A local inclusion principle for U and N is proved by Neil Trudinger and Xu-Jia Wang in ], see the inequality (.3) there. It is then used to show that under the A3 condition a local support function is also global, see ] page 4. The proof is based on a monotone bending argument that gives yet another geometric interpretation of the A3 condition.. Preliminaries.. Optimal transportation. In order to formulate the result in the context of optimal transportation theory we need some standard definitions. Let c : R d R d R be a cost function such that c C 4 (R d R d ) and U, V R d. Definition.. Let u : U R be a continuous function. A c support function of u at x 0 U is ϕ x0 = c(x, y 0 ) + a 0, y 0 R d such that the following two conditions hold u(x 0 ) = ϕ x0 (x 0 ), u(x) ϕ x0 (x), x U. If u has c support at every x 0 U then we say that u is c convex in U. c segment with respect to a point y 0 R d is the set {x R d s.t. c y (x, y 0 ) = line segment}. One may take in the above definition {x R d s.t.c y (x, y 0 ) = tp + ( t)p 0 } with t 0, ] and p 0, p being two points in R d. We say that U is c convex with respect to V R d if the image of the set U under the mapping c y (, y) denoted by c y (U, y) is convex set for all y V. Equivalently, U is c convex with respect to V if for any pair of points x, x U there is y 0 V such that there is a c segment with respect to y 0 joining x with x and lying in U.

3 BLASCHKE S ROLLING BALL THEOREM AND THE TRUDINGER-WANG MONOTONE BENDING 3 Definition.. Let u be a c convex function then the sub-level set of u at x 0 U is (.) S h,u (x 0 ) = {x R d s.t.u(x) < c(x, y 0 ) + u(x 0 ) c(x 0, y 0 )] + h} for some constant h. Equivalently, S h,u (x 0 ) = {x U s.t.u(x) < ϕ x0 (x) + h} where ϕ x0 is the c support function of u at x 0 U, see Definition.. Observe that in the previous definition on may take u(x) = c(x, y ) for some fixed y y 0. Next we recall Kantorovich s formulation of optimal transport problem, see 3, 4]: Let f : U R, g : V R be two nonnegative integrable functions satisfying the mass balance condition f(x)dx = g(y)dy. Then one wishes to minimize (.) u(x)f(x) + U U V V v(y)g(y)dy min among all pairs of functions u : U R, v : V R such that u(x) + v(y) c(x, y). It is well-known that a minimizing pair (u, v) exists 3, 4] and formally the potential u solves the equation f(x) (.3) det(u ij A ij (x, Du)) = detc xi,y j (g y)(x). Here A ij (x, p) = c xi x j (x, y(x, p)) where y(x, p) is determined from D x (c(x, y(x, p))) = p. Assume that c satisfies the following conditions: A For all x, p R d there is unique y = y(x, p) R d such that x c(x, y) = p and for any y, q R d there is unique x = x(y, q) such that y c(x, y) = q. A For all x, y R d detc xi,y j (x, y) 0. A3 For x, p R d there is a positive constant c 0 > 0 such that (.4) A ij,kl (x, p)ξ i ξ j η k η l c 0 ξ η ξ, η R d, ξ η. A3 is the Ma-Trudinger-Wang condition 7]. J.Liu proved that if A-A3 hold then S h,u (x 0 ) is c convex with respect to y 0 6]. There are cost functions satisfying the weak A3 (.5) A ij,kl (x, p)ξ i ξ j η k η l 0 ξ, η R d, ξ η. i.e. when c 0 = 0 in (.4), such that the corresponding sub-level sets are convex in classical sense, see Section 4. We also remark that the condition A3 is equivalent to (.6) d dt c ij(x, y(x, p t ))ξ i ξ j c 0 p p 0 where x is fixed, c x (x, y(x, p t )) = tp + ( t)p 0, t 0, ] c x (x, y) = p, c x (x, y 0 ) = p 0 (this determines the so-called c segment with respect to fixed x), see ].

4 4 ARAM L. KARAKHANYAN.. Shape operator. If M is a surface with positive sectional curvature then by Sacksteder s theorem 0] M is convex. For x M, let n(x) be the unit outward normal at x (n(x) points outside of the convex body bounded by M). The Gauss map x n(x) is a diffeomorphism of M onto S d 5], where S d is the unit sphere in R d. The inverse map n gives a parametrization of M by S d. If M is another smooth convex surface, and w S d, then n (w) and (n ) (w) are the points on M and M with equal outward normals. Let F : Ω R m be a smooth map on a set Ω R d and v = (v,..., v d ) R d then v F (x) = d i= v i F (x) x i is the directional derivative operator. We view the tangent space as a linear subspace of R d consisting of tangential directions. Then the tangent space T x M is the set of vectors perpendicular to n(x). Next we introduce the Weingarten map in order to define the second fundamental form. The Weingarten map W x : T x T x is defined by W x (v) = v n(x). T x is an inner product space (induced by the inner product in R d ). Then W x is self-adjoint operator on T x and the eigenvalues of W x are the principal curvatures at x. Remark.. Observe that since W x is self-adjoint and T x is finite dimensional then there exists an orthonormal basis of T x consisting of eigenvectors of W x. Definition.3. The second fundamental form is defined as II x (v, w) = W x (v) w. When v = w we denote II x (v). From definition it follows that if M is parametrized by r = r(u) and x = r(u 0 ) then (.7) II x (v) = vr n(x), v T x which readily follows from the differentiation of n v r = Main result Theorem 3.. Let y, y V and N (y, y, a) = {x R d : c(x, y 0 ) = c(x, y ) + a} for some a R where c satisfies A, A and weak A3, see (.5). Assume that N is convex for all y, y, a and U is convex domain with smooth boundary such that U is c convex with respect to V, see Definition.. If N and U are internally tangent at some point z 0 then U is inside N. Using the terminology of Blaschke s theorem it follows that under the conditions of Theorem 3. U rolls freely inside N. Observe that the c convexity of sub-level sets is known under stronger condition A3 6]. In the next section we give an example of cost function c satisfying weaker form of A3 (.5) but such that N is convex for all y, y, a. Proof to follow is inspired in ].

5 BLASCHKE S ROLLING BALL THEOREM AND THE TRUDINGER-WANG MONOTONE BENDING 5 Proof. Step : (Parametrizations) To apply Theorem. we take M = U and M = N and assume that U and N are internally tangent at z 0. Assume that at x 0 N and x 0 U N and U have the same outward normal, see Figure. In what follows we use the following radial parametrizations: U R(ζ), ζ D U, N X(ω), ω D N, (c y ( U, y 0 )) ρ(ζ) = c y (R(ζ), y 0 ). Here D U and D N are the domains of corresponding parameters. Moreover, there are ω D N and ζ D U such that (3.) x 0 := X( ω) N and x 0 := R( ζ) U. From now on ζ and ω are fixed. Let n( ζ) denote the outward normal of the image c y (U, y 0 ) at the point ρ( ζ). We have (3.) n m ( ζ) = c ym,x i (R( ζ), y 0 )n i ( ζ). Observe that by assumption the constant matrix µ = c ym,x i (R( ζ), y 0 )] has non-trivial determinant, see A. Furthermore, the set µc y (U, y 0 ) = {µx s.t. x c y (U, y 0 )} is again convex because for any two points q = µz, q = µz such that q, q µc y (U, y 0 ) and z, z c y (U, y 0 ) we have for all θ 0, ]. µc y (U, y 0 ) µ(θz + ( θ)z ) = θµz + ( θ)µz = θq + ( θ)q Step : (Computing the second fundamental form of X) Next, we introduce the vectorfield r = r(ζ), ζ D U such that (3.3) r(ζ) = µρ(ζ) = µc y (R(ζ), y 0 ). We compute the first and second derivatives (3.4) (3.5) rζ m s := rs m = µ αβ c yβ,x i Rs, i ] rst m = µ αβ c yβ,x i x j RsR i j t + c y β,x i Rst i. From (3.4) and (3.) we see that at r( ζ) the normal is (3.6) n( ζ) = µ n( ζ). Take p t = ( t)p 0 + tp, t 0, ] and (3.7) p t = c x (x 0, y(x 0, p t )),

6 6 ARAM L. KARAKHANYAN Figure. Schematic view to parametrizations of U, N, (c y (U, y 0 )) and µ (c y (U, y 0 )). then y t := y(x 0, p t) defines the c segment joining y 0 and y, see A. In particular, one has (3.8) p i p i 0 = c xi,y m (x 0, y(x 0, p t )) d dt ym (x 0, p t ) = d dt ym (x 0, p t )c ym,x i (x 0, y(x 0, p t )). Let X t (ω) be the parametrization of N (t) = {x U : c(x, y 0 ) = c(x, y t ) + a} (recall that N (t) is convex as the boundary of sub-level set). We can choose a = a(t) so that all N (t) pass through the point x 0, in other words there is ωt such that X t ( ω t ) = x 0. Moreover, by (3.7) it follows that (3.9) c xi (X t ( ω t ), y 0 ) c xi (X t ( ω t ), y t ) = c xi (x 0, y 0 ) c xi (x 0, y t ) = p i 0 p i t = t(p i 0 p i ). After fixing t and differentiating the identity c(x t (ω), y 0 ) = c(x t (ω), y t ) + a(t) in ω we get cxi (X t, y 0 ) c xi (X t, y t ) ] X i,t = 0, (3.0) cxi x j (X t, y 0 ) c xi x j (X t, y t ) ] X j,t ω l X i,t + c xi (X t, y 0 ) c xi (X t, y t ) ] X i,t = 0. Thus the normals of N (t) at x 0 are collinear to p p 0 for all t 0, ], that is

7 BLASCHKE S ROLLING BALL THEOREM AND THE TRUDINGER-WANG MONOTONE BENDING 7 (3.) n(x 0 ) = n (x 0) = p p 0, µ n = n (recall (3.6)). p p 0 Hence we can rewrite (3.0) as follows (3.) (cxi x j (X t, y 0 ) c xi x j (X t, y t ) ] X j,t ω l X i,t = t(p i 0 p i )X i,t. Keeping X t ( ω t ) = x 0 fixed for all t 0, ], dividing both sides of the last identity by t and then sending t 0 we obtain (3.3) y (x 0, p 0 )c y,xi x j (x 0, y 0 ) ] X j,t=0 ω l X i,t=0 = (p i 0 p i )X i,t=0. On the other hand from (3.8) we see that d dt y(x 0, p t) t=0 = (p p 0 )µ. Thus substituting this into the last equality we obtain (3.4) (p p 0 )µc y,xi x j (x 0, y 0 ) ] X j,t=0 ω l X i,t=0 = (p i 0 p i )X i,t=0 = (p i p i 0)X i,t=0 or equivalently (3.5) n α µ αβ c yβ,x i x j (x 0, y 0 ) ] X j,t=0 ω l X i,t=0 = n i X i,t=0 if we utilize (3.). Step 3: (Monotone bending) By Remark. we assume that T x0 U and T x 0 N (t = 0) have the same local coordinate system (by reparametrizing N (t = 0) if necessary). From convexity of µc y (U, y 0 ) boundary of which is parametrized by r we have (3.6) 0 r α stn α = µρ st n = µ αβ (c yβ,x i x j R i sr j t + c y β,x i R i st)n α = µ αβ c yβ,x i x j R i sr j t nα + R i stn i (3.5) = n i X i,t=0 + R i stn i. Now (.6) yields that at x 0 (3.7) n i X i,t n i X i,t=0 (3.6) R i stn i. Recalling (.7) we finally obtain the required inequality II x 0 N II x0 U. The proof is now complete. Note that weak A3 (i.e. when c 0 = 0 in (.6)) is enough for the monotonicity to conclude the inequality n i X i,t n i Xω i,t=0 k ω l.

8 8 ARAM L. KARAKHANYAN 4. Applications 4.. Convex sub-level sets. There is a wide class of cost functions for which the set N is convex. Observe that c(x, y) = p x y p satisfies A3 for < p < and weak A3 if p = ± 7]. It is useful to note that if Ω ψ = {x R d s.t. ψ(x) < 0} for some smooth function ψ : R d R such that Ω ψ then (4.) ψ(x)τ(x) τ(x) 0, τ(x) T x is a necessary and sufficient condition for Ω ψ to be convex provided that ψ ψ is directed towards positive ψ. Using this, one can check that the boundaries of sub-level sets (e.g. p = ) x y x y a = 0 are convex. Here ψ(x) := x y x y a < 0 defines the sub-level set. Indeed, using (4.) we see that for any unit vector τ perpendicular to the vector ψ(x) = (x y ) x y 4 + (x y ) x y 4 we get ττ ψ = x y 4 4((x y ) τ) x y ψ τ=0 = = = = ] + x y 4 4((x y ) τ) ] x y x y 4 x y 4 8((x y ) τ) x y 6 + 8((x y ) τ) x y 8 x y 6 x y 8 x y 4 x y 4 + 8((x y ) τ) x y ] x y 6 x y x y 4 x y 4 Altogether, we infer that ( ) x y + a + 8((x y ) τ) x y 6 ( ) x y + a 8((x y ) τ) ττ ψ 0 if a 0. x y x y + a a x y 6 x y + a. If a < 0 then we can swap y and y to conclude that ττ ψ > 0. Some examples of sub-level sets of inverse quadratic cost function are illustrated in Figure. 4.. Antenna design problems. In parallel reflector problem 4] one deals with the paraboloids of revolution (4.) P (x, σ, Z) = σ + Zn+ x z σ ]

9 .6 BLASCHKE S ROLLING BALL THEOREM AND THE TRUDINGER-WANG MONOTONE BENDING Figure. From left to right: a =, y = ( 0 3, 0), y = (, 0 ); a =, y = ( 0, 0 ), y = (, 0 ); a =, y = ( 0 4, 0), y = (., 0 ). which play the role of support functions. Here the point Z = (z, Z n+ ) R n+ is the focus of the paraboloid such that ψ(z, Z n+ ) = 0 for some smooth function ψ satisfying some structural conditions and σ is a constant. If P is internally tangent to P at z 0 and II z0 P II z0 P then P is inside P, see Lemma 8. 4]. This again follows from Blaschke s theorem. Indeed, we have that at the points x and x corresponding to coinciding outward normals and II x P = II x P = Furthermore DP (x) = DP (x ) and hence δ ij + DP (x) σ δ ij. + DP (x ) σ (4.3) + DP (x) = + DP (x ). From II z0 P II z0 P we infer that (4.4) Consequently (4.4) and (4.3) imply that σ σ. II x P II x P Another inclusion principle. There are various inclusion principles in geometry, we want to mention the following elementary one due to Nitsche 8]: Each continuous closed curve of length L in Euclidean 3-space is contained in a closed ball of radius R < L/4. Equality holds only for a needle, i.e., a segment of length L/ gone through twice, in opposite directions. Later J. Spruck generalized this result for compact Riemannian manifold M of dimension n 3 as follows: if the sectional curvatures K(σ) /c for

10 0 ARAM L. KARAKHANYAN all tangent plane sections σ then M is contained in a ball of radius R < πc, and this bound is best possible. We remark here that there is a smooth surface S R 3 such that the mean curvature H and the Gauss curvature K then the unit ball cannot be fit inside S, see ]. Notice that K is an intrinsic quantity and H implies that K. References ] W. Blaschke, Kreis und Kugel, 3rd edition. Berlin, de Gruyter, 956 ] J. N. Brooks, J. B. Strantzen, Blaschke s rolling theorem in R n. Mem. Amer. Math. Soc. 80 (989), no ] J.A. Delgado, Blaschke s theorem for convex hypersurfaces. J. Differential Geom. 4 (979), no. 4, ] A.L. Karakhanyan, Existence and regularity of the reflector surfaces in R n+. Arch. Ration. Mech. Anal. 3 (04), no. 3, ] D. Koutroufiotis, On Blaschke s rolling theorems, Arch. Math. 8 (97) ] J. Liu, Hölder regularity of optimal mappings in optimal transportation. Calc. Var. Partial Differential Equations 34 (009), no. 4, ] X.N. Ma, N.S. Trudinger, X.-J. Wang, Regularity of potential functions of the optimal transportation problem. Arch. Rat. Mech. Anal. 77, 5?83 (005) 8] J. C. C. Nitsche, The smallest sphere containing a rectifiable curve, Amer. Math. Monthly 7 (97) ] J. Rauch, An inclusion theorem for ovaloids with comparable second fundamental forms. J. Differential Geometry 9 (974), ] R. Sacksteder, On hypersurfaces with non-negative sectional curvatures, Amer. J. Math. 8 (960) ] J. Spruck, On the radius of the smallest ball containing a compact manifold of positive curvature. J. Differential Geometry 8 (973), ] N. Trudinger, X.-J. Wang, On strict convexity and continuous differentiability of Potential functions in optimal transportation, Arch Rat. Mech. Anal ] J. Urbas, Mass transfer problems, Lecture Notes, Univ. of Bonn, 998

11 BLASCHKE S ROLLING BALL THEOREM AND THE TRUDINGER-WANG MONOTONE BENDING 4] C. Villani, Optimal transport, old and new. Notes for the 005 Saint-Flour summer school. Grundlehren der mathematischen Wissenschaften, Springer 008 5] H. Wu, The Spherical images of convex hypersurfaces, J. Differential Geometry 9 (974) Aram L. Karakhanyan, School of Mathematics, The University of Edinburgh, Mayfield Road, EH9 3JZ, Edinburgh, UK address: aram.karakhanyan@ed.ac.uk