Regularity of shadows and the geometry of the singular set associated to a Monge-Ampere equation
RREGULARITY OF SHADOWS AND THE GEOMETRY OF THESINGULAR SET ASSOCIATED TO A MONGE-AMP `ERE EQUATION
E. INDREI AND L. NURBEKYAN
Abstract.
Illuminating the surface of a convex body with parallel beams of light ina given direction generates a shadow region. We prove sharp regularity results for theboundary of this shadow in every direction of illumination. Moreover, techniques aredeveloped for investigating the regularity of the region generated by orthogonally pro-jecting a convex set onto another. As an application we study the geometry and Hausdorffdimension of the singular set corresponding to a Monge-Amp`ere equation. Introduction
Shadows play an important role in many different branches of mathematics such asdifferential geometry, convex analysis, geometric combinatorics, and functional analysis[11, 6, 3, 12, 20, 2, 17, 1]. Our aim in this paper is to show that they also naturallyappear in a free boundary problem associated to a Monge-Amp`ere equation. Indeed, itturns out that the regularity of certain shadow regions yields information on the Hausdorffdimension of the singular set appearing in the optimal partial transport problem [7, 9,10, 16, 8].1.1.
Illumination shadows.
Illumination shadows form powerful tools in the classifi-cation of surfaces. For instance, it is a well-known fact that if every shadow boundarygenerated by parallel illumination on a Blaschke surface embedded in R is a plane curve,then the surface is quadric [19]. Moreover, in 2001 Ghomi [11] solved the shadow prob-lem formulated in 1978 by Wente: if M is a closed oriented 2-dimensional manifold and f : M → R is a smooth immersion, then f is a convex embedding if and only if theshadow region generated by parallel illumination is simply connected in every direction.Regularity properties of shadow boundaries have been investigated in [11, 23, 12, 24,14, 15]. For example, given a smooth manifold it is well known that if the Gaussiancurvature does not vanish at a given point, then the shadow boundary is locally smootharound that point (via the inverse function theorem). Moreover, using Sard’s theorem,it is not difficult to prove that in almost every direction (in the sense of Lebesgue), the The first author acknowledges support from the Australian Research Council and US NSF grantDMS-0932078, administered by the Mathematical Sciences Research Institute in Berkeley, California.The second author acknowledges support from the department of mathematics at the University of Texasat Austin and the Centre for Mathematical Analysis, Geometry, and Dynamical Systems at InstitutoSuperior T´ecnico. a r X i v : . [ m a t h . A P ] N ov E. INDREI AND L. NURBEKYAN shadow boundary of a surface is continuous. Steenarts [23] showed that for a smoothconvex body, the shadow boundary has finite ( n − G × S n − , most pairs (Ω , u ) ∈ G × S n − (i.e. up to a meagre set in the sense of Baire)generate boundaries with infinite ( n − n − §
2, we address this problem with tools from convex analysis and obtain the followingresults. • [Theorem 2.4] For a strictly convex domain in R n , the boundary of the shadowgenerated by parallel illumination is locally a continuous graph in every direction. • [ § • [Theorem 2.5] For a uniformly convex C ,α domain in R n , α ∈ (0 , C ,α graph in everydirection. • [ § α ∈ (0 , C ∞ smooth convex set and a direction, sothat the shadow boundary generated by parallel illumination in that direction is C β for β < α . In particular, one may not remove the uniform convexity assumptionin Theorem 2.5. • [Remark 2.7] For a uniformly convex C k +1 domain in R n , k ≥
1, the boundary ofthe shadow generated by parallel illumination is locally a C k graph.We note that shadows generated by another type of illumination process also appearin a well-known covering problem of Levi [18] and Hadwiger [13]: let Ω ⊂ R n be a convexbody and h (Ω) the smallest number so that Ω can be covered by h (Ω) smaller homotheticalcopies of itself; the conjecture states that h (Ω) ≤ n , with equality if and only if Ω is an n -parallelotope. Indeed, Boltyanskii [5] connected this conjecture with an illuminationproblem by showing that h (Ω) = l (Ω) where l (Ω) is the smallest number of light sourcesoutside of Ω required to illuminate ∂ Ω; a boundary point y of Ω is said to be illuminatedfrom x / ∈ Ω if the line through x and y intersects the interior of Ω. For further reading,we refer the reader to two survey articles [2, 20] and the references therein. Projection shadows.
In 1986 Kiselman [17] addressed the following question: whatdegree of smoothness does a two-dimensional projection of a three-dimensional smoothconvex set possess? He proved that if the convex set is C , then its projection is also C ; if it is C , , then the boundary of the projection is twice differentiable; and, if it isreal-analytic, then the boundary of the projection is C ,α for some α >
0. Moreover, heprovided examples to show that these results are essentially sharp: in the real-analyticcase, the boundary of the projection may be exactly C , q for any odd integer q ≥ C ∞ set may not be C [17, Example3.3]. V. Sedykh [22] studied this question in higher dimensions and proved that theprojection of a smooth closed convex surface in R n onto a hyperplane is C , and showedthat this result is sharp in the sense that there exists a hypersurface whose shadow is nottwice differentiable; this contrasts with Kiselman’s result in R . Moreover, the analyticcase also displays a loss of regularity in higher dimensions: Bogaevsky [4] showed theexistence of a real-analytic closed convex hypersurface, whose shadow does not belong tothe class C . These results are all compiled and discussed in the book “Arnold’s problems”by V.I. Arnold [1] (Arnold calls these types of shadows “apparent contours”).In applications, however, one may require regularity results of this shadow when pro-jecting onto a strictly convex domain (as opposed to a hyperplane as in the results above),see e.g. § ⊂ R n , Λ ⊂ R n , if P Λ (Ω) denotes the orthogonalprojection of Ω onto Λ, then how smooth is ∂ ( P Λ (Ω) ∩ ∂ Λ)? The following results areestablished in § • [Theorem 3.1] Let Ω ⊂ R n be a bounded strictly convex domain and Λ ⊂ R n aconvex domain whose boundary is C , . If Ω ∩ Λ = ∅ , then ∂P Λ (Ω) is finitely( n − • [Remark 3.3] The disjointness assumption in Theorem 3.1 is necessary: there existtwo bounded convex domains Ω and Λ in R for which H ( ∂ ( P Λ (Ω) ∩ ∂ Λ)) = ∞ . • [Theorem 3.4] If Ω and Λ are C k +1 convex domains in R n with disjoint closures, k ≥
1, and Ω is bounded and uniformly convex, then ∂P Λ (Ω) is an ( n − C kloc hypersurface.We point out that when one takes Λ to be a hyperplane, Theorem 3.1 is immediate:the projection of a convex set onto a hyperplane is convex, so ∂P Λ (Ω) is locally Lipschitz.However, the situation is different if Λ is curved. Here is the idea of our method: wetake a point y ∈ ∂P Λ (Ω) and represent Λ locally by a bi-Lipschitz graph with respect E. INDREI AND L. NURBEKYAN to the tangent space at y , T y Λ = R n − . Then we consider P R n − ∂P Λ (Ω) and cook upan auxiliary uniformly convex C , function that touches this set at P R n − ∂P Λ ( y ). Byapplying our results from Theorem 2.5 (or rather, the idea in the proof), we show thatthere exists a Lipschitz function which touches P R n − ∂P Λ (Ω) at P R n − ∂P Λ ( y ) and bounds P R n − ∂P Λ (Ω) from one side (in a suitable coordinate system). This yields the existenceof a cone whose opening can be shown to depend only on the initial data (i.e. Ω andΛ) and that touches P R n − ∂P Λ (Ω) only at P R n − ∂P Λ ( y ); the rest follows by iterating theargument above and locally transporting cones at all the other points in P R n − ∂P Λ (Ω)from the surrounding tangent spaces via the C , charts representing Λ and applying astandard covering argument from geometric measure theory.The idea of this argument in terms of finding a cone was employed by Indrei [16],although he assumed Λ to be uniformly convex. The novelty in this paper is that weconstruct our barrier-type function without requiring uniform convexity of Λ. Indeed,this support function is constructed by using the boundary of the shadow generated byilluminating Λ in the direction of some normal of Ω at the point y + t ( y ) N Λ ( y ), where t ( y )is the first hitting time of Ω. However, in contrast with [16, Proposition 4.9], we requirea strict convexity assumption on Ω. Nevertheless, this tradeoff turns out to be moreuseful when applying our theory to a free boundary problem that has a strict convexityassumption on Ω naturally built into it, see § ∂ Ω and ∂ Λ locally as level sets of twoconvex functions functions G : R n → R and F : R n → R . By exploiting the geometry ofthe problem, we construct a function φ : R n +1 → R n +3 so that ∂P Λ (Ω) is locally a levelset of φ (herein lies the novelty of our approach since we are connecting the two sets andthe unknown shadow boundary by a single function); next, we compute the differential ofthis map and show that it has full rank and conclude via the implicit function theorem.1.3. Shadows and a Monge-Amp`ere equation.
The optimal partial transport prob-lem is a generalization of the classical Monge-Kantorovich problem: given two non-negative functions f = f χ Ω , g = gχ Λ ∈ L ( R n ) and a number 0 < m ≤ min {|| f || L , || g || L } , the objective is to find an optimal transference plan between f and g with mass m . Atransference plan is a non-negative, finite Borel measure γ on R n × R n , whose first andsecond marginals are controlled by f and g respectively: for any Borel set A ⊂ R n , γ ( A × R n ) ≤ (cid:90) A f ( x ) dx, γ ( R n × A ) ≤ (cid:90) A g ( x ) dx. An optimal transference plan is a minimizer of the functional(1.1) γ → (cid:90) R n × R n c ( x, y ) dγ ( x, y ) , where c is a non-negative cost function. Issues of existence, uniqueness, and regularityof optimal transference plans have been addressed by Caffarelli & McCann [7], Figalli[9, 10], Indrei [16], and Chen & Indrei [8]. If || f ∧ g || L ( R n ) ≤ m ≤ min {|| f || L ( R n ) , || g || L ( R n ) } , then by the results in [9, Section 2], there exists a convex function Ψ m and non-negativefunctions f m , g m for which γ m := ( Id × ∇ Ψ m ) f m = ( ∇ Ψ ∗ m × Id ) g m , is the unique solution of (1.1) and ∇ Ψ m f m = g m (see [9, Theorem 2.3]).Ψ m is known as the Brenier solution of the Monge-Amp`ere equationdet( D Ψ m )( x ) = f m ( x ) g m ( ∇ Ψ m ( x )) , with x ∈ F m := set of density points of { f m > } , and ∇ Ψ m ( F m ) ⊂ G m := set of densitypoints of { g m > } . Moreover, as in [9, Remark 3.2], we set U m := (Ω ∩ Λ) ∪ (cid:91) (¯ x, ¯ y ) ∈ Γ m B | ¯ x − ¯ y | (¯ y ) ,V m := (Ω ∩ Λ) ∪ (cid:91) (¯ x, ¯ y ) ∈ Γ m B | ¯ x − ¯ y | (¯ x ) , where Γ m is the set( Id × ∇ Ψ m )( F m ∩ D ∇ Ψ m ) ∩ ( ∇ Ψ ∗ m × Id )( G m ∩ D ∇ Ψ ∗ m ) , with D ∇ Ψ m and D ∇ Ψ ∗ m denoting the set of continuity points for ∇ Ψ m and ∇ Ψ ∗ m , respec-tively, where Ψ ∗ m is the Legendre transform of Ψ m .The free boundary associated to f m is denoted by ∂U m ∩ Ω and the free boundaryassociated to g m by ∂V m ∩ Λ. They correspond to ∂F m ∩ Ω and ∂G m ∩ Λ, respectively[9, Remark 3.3]. One method of obtaining free boundary regularity is to first proveregularity results on Ψ m and then utilize that ∇ Ψ m gives the direction of the normal tothe free boundary ∂U m ∩ Ω (cid:0) by symmetry and duality, this also implies a similar resultfor ∂V m ∩ Λ (cid:1) .Indeed, this method was employed by Caffarelli & McCann [7] to deduce C ,αloc freeboundary regularity away from a singular set ˜ S in the case when Ω and Λ are strictlyconvex and separated by a hyperplane. Indrei [16] generalized an improvement of thisresult in the overlapping case: he obtains C ,αloc free boundary regularity away from thecommon region Ω ∩ Λ and a singular set S which in the disjoint case is a subset of ˜ S .Moreover, he developed a method to study the Hausdorff dimension of ˜ S and utilized it toprove that if the domains are C , and uniformly convex, then S has Hausdorff dimension( n − §
4, we connect the shadow boundaries with this singular set and show that onemay replace the uniform convexity assumption with a strict convexity assumption toobtain that the singular set has Hausdorff dimension ( n − E. INDREI AND L. NURBEKYAN be handled using notions from transport theory and non-smooth analysis; the other canbe shown to be trapped on the boundary of P Λ (Ω). Thus, understanding the Hausdorffdimension of the boundary of this shadow is a way to obtain bounds on the Hausdorffdimension of the singular set. This is where the rectifiability result of Theorem 3.1 comesinto play. Since Theorem 2.5 was used in the proof of Theorem 3.1, this highlights theinterplay between the shadow generated by parallel illumination, the shadow generated byorthogonal projections, and the Monge-Amp`ere free boundary problem arising in optimaltransport theory.2. Regularity of shadows generated by parallel illumination
In this section we investigate the regularity of the shadow region of a convex domainΛ ⊂ R n under parallel illumination. For u ∈ S n − , we denote the shadow of Λ by theset S u of points x ∈ ∂ Λ such that there exists a normal vector ν ( x ) (i.e. a vector in thenormal cone of Λ at x ) for which (cid:104) ν ( x ) , u (cid:105) >
0. Our aim is to prove that for a strictlyconvex domain Λ, the boundary ∂S u (in the topology of ∂ Λ) is locally a continuous graphand that this regularity is optimal in the sense that if Λ is not strictly convex, then ∂S u might fail to locally be a graph.Given k ∈ N and x = ( x , x , · · · , x k ) ∈ R k , we denote an arbitrary vector in R k − by x (cid:48) := ( x , x , · · · , x k − ) . Furthermore, let x (cid:48)(cid:48) := ( x (cid:48) ) (cid:48) = ( x , x , · · · , x k − ) ∈ R k − . For aset A ⊂ R k define A (cid:48) := { x (cid:48) : x ∈ A } and A (cid:48)(cid:48) := ( A (cid:48) ) (cid:48) .Let x ∈ ∂ Λ be a boundary point of the shadow S u . Without loss of generality we mayassume that in a neighborhood of x , say U , Λ is parametrized as x n ≤ φ ( x (cid:48) ), for somestrictly concave function φ . Consequently, ∂ Λ is locally given by x n = φ ( x (cid:48) ) where thedomain of φ is U (cid:48) ⊂ R n − . Note that x (cid:48) (cid:55)→ ( x (cid:48) , φ ( x (cid:48) )) is a homeomorphism between thespaces U (cid:48) and ∂ Λ ∩ U .For every y (cid:48) ∈ U (cid:48) there is a one-to-one correspondence between superdifferentials w ∈ ∂ + φ ( y (cid:48) ) and normals ν at ( y (cid:48) , φ ( y (cid:48) )) given by ν = ( − w, | w | +1) / . Therefore ( y (cid:48) , φ ( y (cid:48) )) ∈ S u ifand only if (cid:104) w ( y (cid:48) ) , u (cid:48) (cid:105) < u n , for some w ∈ ∂ + φ ( y (cid:48) ).In this section we prove that S (cid:48) u ∩ U (cid:48) (in the usual R n − topology) is locally a continuousgraph. By rotating the coordinate system, if necessary, we may assume x = 0 and u (cid:48) =(0 , , · · · , ∈ R n − ; moreover, we identify R n − with ( u (cid:48) ) ⊥ . Under these assumptions,the condition (cid:104) w, u (cid:48) (cid:105) < u n takes the form w n − < u n . We begin our analysis with thefollowing lemma. Lemma 2.1.
Let Λ ⊂ R n be a strictly convex domain and y (cid:48) ∈ S (cid:48) u ∩ U (cid:48) . Then ( y (cid:48)(cid:48) , α ) ∈ S (cid:48) u ,for every α > y n − such that ( y (cid:48)(cid:48) , α ) ∈ U (cid:48) .Proof. Since y (cid:48) ∈ S (cid:48) u , there exists w ∈ ∂ + φ ( y (cid:48) ) such that w n − < u n . Let w ∈ ∂ + φ ( y (cid:48)(cid:48) , α )be any element in the superdifferential. By the monotonicity formula, (cid:104) w − w , ( y (cid:48)(cid:48) , α ) − y (cid:48) (cid:105) < , or equivalently ( w n − − w n − )( α − y n − ) <
0. Therefore, w n − < w n − . Combining thiswith w n − < u n yields w n − < u n , and this implies ( y (cid:48)(cid:48) , α ) ∈ S (cid:48) u . (cid:3) Lemma 2.2.
Let Λ ⊂ R n be a strictly convex domain. Then there exists a ball V (cid:48)(cid:48) ⊂ R n − centered at (cid:48)(cid:48) with the following properties: for every y (cid:48)(cid:48) ∈ V (cid:48)(cid:48) , there exist α, β ∈ R suchthat ( y (cid:48)(cid:48) , α ) , ( y (cid:48)(cid:48) , β ) ∈ U (cid:48) and for every η ∈ ∂ + φ ( y (cid:48)(cid:48) , α ) and ζ ∈ ∂ + φ ( y (cid:48)(cid:48) , β ) , one has η n − < u n < ζ n − .Proof. Since the set ∂ + φ (0 (cid:48) ) is convex, one of the following is true:(i) w n − < u n for every w ∈ ∂ + φ (0 (cid:48) );(ii) w n − > u n for every w ∈ ∂ + φ (0 (cid:48) );(iii) w n − = u n for some w ∈ ∂ + φ (0 (cid:48) ).However, by continuity properties of the superdifferential of a convex function (see e.g.[21, Corollary 24.5.1]), if (i) or (ii) holds, then the strict inequality will be satisfied in someneighborhood of 0 (cid:48) . This contradicts 0 (cid:48) ∈ ∂S (cid:48) u . Hence, w n − = u n for some w ∈ ∂ + φ (0 (cid:48) ).Pick β < < α such that (0 (cid:48)(cid:48) , α ) , (0 (cid:48)(cid:48) , β ) ∈ U (cid:48) . The monotonicity formula implies thatevery η ∈ ∂ + φ (0 (cid:48)(cid:48) , α ) satisfies η n − < w n − = u n , and similarly every ζ ∈ ∂ + φ (0 (cid:48)(cid:48) , β )satisfies ζ n − > w n − = u n . By utilizing the continuity of the superdifferential again,there exists a ball V (cid:48)(cid:48) centered at 0 (cid:48)(cid:48) such that for every y (cid:48)(cid:48) ∈ V (cid:48)(cid:48) with ( y (cid:48)(cid:48) , α ) , ( y (cid:48)(cid:48) , β ) ∈ U (cid:48) ,we have that for every η ∈ ∂ + φ ( y (cid:48)(cid:48) , α ) and ζ ∈ ∂ + φ ( y (cid:48)(cid:48) , β ), η n − < u n < ζ n − . (cid:3) Let V (cid:48)(cid:48) be the ball from Lemma 2.2. For every y (cid:48)(cid:48) ∈ V (cid:48)(cid:48) , define(2.1) γ ( y (cid:48)(cid:48) ) := inf { t : ( y (cid:48)(cid:48) , t ) ∈ S (cid:48) u ∩ U (cid:48) } . By Lemma 2.2, γ is well-defined with ( y (cid:48)(cid:48) , γ ( y (cid:48)(cid:48) )) ∈ U (cid:48) . Lemma 2.3. (Properties of γ ) Let Λ ⊂ R n be a strictly convex domain and V (cid:48)(cid:48) the ballfrom Lemma 2.2. For every y (cid:48)(cid:48) ∈ V (cid:48)(cid:48) there exists w ∈ ∂ + φ ( y (cid:48)(cid:48) , γ ( y (cid:48)(cid:48) )) such that w n − = u n .Moreover, if t > γ ( y (cid:48)(cid:48) ) and ( y (cid:48)(cid:48) , t ) ∈ U (cid:48) , then ζ n − < u n for every ζ ∈ ∂ + φ ( y (cid:48)(cid:48) , t ) . Similarly,if t < γ ( y (cid:48)(cid:48) ) and ( y (cid:48)(cid:48) , t ) ∈ U (cid:48) , then ζ n − > u n for every ζ ∈ ∂ + φ ( y (cid:48)(cid:48) , t ) . Furthermore, γ iscontinuous.Proof. Suppose w n − < u n for all w ∈ ∂ + φ ( y (cid:48)(cid:48) , γ ( y (cid:48)(cid:48) )). By continuity of the superdif-ferential, ζ n − < u n for all ζ ∈ ∂ + φ ( y (cid:48)(cid:48) , t ) if t is sufficiently close to γ ( y (cid:48)(cid:48) ); therefore,( y (cid:48)(cid:48) , t ) ∈ S (cid:48) u ∩ U (cid:48) for some t < γ ( y (cid:48)(cid:48) ), and this contradicts the definition of γ . On the otherhand, if w n − > u n for all w ∈ ∂ + φ ( y (cid:48)(cid:48) , γ ( y (cid:48)(cid:48) )), then again by continuity of the superdif-ferential, ζ n − > u n for all ζ ∈ ∂ + φ ( y (cid:48)(cid:48) , t ) if t is sufficiently close to γ ( y (cid:48)(cid:48) ). This implies( y (cid:48)(cid:48) , t ) ∈ ( S (cid:48) u ) c ∩ U (cid:48) for γ ( y (cid:48)(cid:48) ) < t ≤ t , with t sufficiently close to γ ( y (cid:48)(cid:48) ); again, this pro-duces a contradiction. Therefore, there exist w , w ∈ ∂ + φ ( y (cid:48)(cid:48) , γ ( y (cid:48)(cid:48) )) such that w n − ≤ u n and w n − ≥ u n . Hence, for some s ∈ [0 , w n − = u n where w = (1 − s ) w + sw . Since ∂ + φ ( y (cid:48)(cid:48) , γ ( y (cid:48)(cid:48) )) is a convex set, it follows that w ∈ ∂ + φ ( y (cid:48)(cid:48) , γ ( y (cid:48)(cid:48) )). Now pick any t > γ ( y (cid:48)(cid:48) ).By the monotonicity formula, for every ζ ∈ ∂ + φ ( y (cid:48)(cid:48) , t ), ζ n − < w n − = u n . Similarly, if t < γ ( y (cid:48)(cid:48) ), then η n − > w n − = u n for all η ∈ ∂ + φ ( y (cid:48)(cid:48) , t ). Next let y (cid:48)(cid:48) k , y (cid:48)(cid:48) ∈ V (cid:48)(cid:48) and y (cid:48)(cid:48) k → y (cid:48)(cid:48) .Since { γ ( y (cid:48)(cid:48) k ) } is a bounded sequence, every subsequence has a further subsequence thatconverges. Take such a subsequence and suppose it converges to, say, t ∈ R . If t > γ ( y (cid:48)(cid:48) ),then by what has already been proved, we have that ζ n − < u n for all ζ ∈ ∂ + φ ( y (cid:48)(cid:48) , t ).Therefore, by the continuity of the superdifferential, this condition is satisfied in some E. INDREI AND L. NURBEKYAN neighborhood of ( y (cid:48)(cid:48) , t ), but this contradicts the fact that ( y (cid:48)(cid:48) k , γ ( y (cid:48)(cid:48) k )) → ( y (cid:48)(cid:48) , t ) (alongthis subsequence) and that there exists w k ∈ ∂ + φ ( y (cid:48)(cid:48) k , γ ( y (cid:48)(cid:48) k )) such that w kn − = u n . Thecase t < γ ( y (cid:48)(cid:48) ) may be excluded in the same manner. Hence, t = γ ( y (cid:48)(cid:48) ), and we provedthat every subsequence of { γ ( y (cid:48)(cid:48) k ) } admits a further subsequence converging to γ ( y (cid:48)(cid:48) ); thisimplies the continuity of γ . (cid:3) Now we have all the ingredients to prove the following theorem which may be seen asthe first step towards investigating the regularity of the boundary of the shadow region.
Theorem 2.4.
Let Λ ⊂ R n be a strictly convex domain and u ∈ S n − . Then the boundaryof the shadow region generated by parallel illumination in the direction u is locally the graphof a continuous function. More precisely, (2.2) ∂S (cid:48) u ∩ U (cid:48) ∩ ( V (cid:48)(cid:48) × R ) = { ( y (cid:48)(cid:48) , γ ( y (cid:48)(cid:48) )) : y (cid:48)(cid:48) ∈ V (cid:48)(cid:48) } . Proof.
Lemma 2.3 implies the continuity of γ and that ( y (cid:48)(cid:48) , α ) ∈ S (cid:48) u for every α > γ ( y (cid:48)(cid:48) )and ( y (cid:48)(cid:48) , α ) ∈ ( S (cid:48) u ) c for every α < γ ( y (cid:48)(cid:48) ), where ( y (cid:48)(cid:48) , α ) ∈ U (cid:48) ∩ ( V (cid:48)(cid:48) × R ). (cid:3) If the convex domain to be illuminated is uniformly convex, then the shadow boundary islocally H¨older continuous under mild regularity assumptions. The next theorem quantifiesthis statement.
Theorem 2.5. If Λ ⊂ R n is a uniformly convex C ,α domain, α ∈ (0 , , then ∂S (cid:48) u islocally a C ,α graph.Proof. From Theorem 2.4 it follows that ∂S (cid:48) u is the graph of a continuous function γ defined on the ball V (cid:48)(cid:48) . Therefore, it suffices to show that γ is H¨older continuous on V (cid:48)(cid:48) .Lemma 2.3 implies that for every y (cid:48)(cid:48) ∈ V (cid:48)(cid:48) , γ ( y (cid:48)(cid:48) ) is the only solution of the equation ∂φ∂y n − ( y (cid:48)(cid:48) , α ) = u n , hence(2.3) ∂φ∂y n − ( y (cid:48)(cid:48) , γ ( y (cid:48)(cid:48) )) = u n , (recall that φ is the local chart representing ∂ Λ). Since Λ is C ,α and uniformly convex, φ is C ,α and uniformly concave, i.e.(2.4) |∇ φ ( y (cid:48) ) − ∇ φ ( z (cid:48) ) | ≤ L | y (cid:48) − z (cid:48) | α and(2.5) (cid:104)∇ φ ( y (cid:48) ) − ∇ φ ( z (cid:48) ) , y (cid:48) − z (cid:48) (cid:105) ≤ − θ | y (cid:48) − z (cid:48) | , for some L, θ > y (cid:48) , z (cid:48) ∈ V (cid:48) . To prove that γ is H¨older, it suffices to show thatat every point on the graph of γ , we can place a cusp with uniform opening that staysabove the graph. It suffices to prove it for one point since the proof is identical for anyother point. Without loss of generality, we assume 0 (cid:48) ∈ ∂S (cid:48) u and show that a cusp can beplaced at 0 (cid:48) that stays above the graph: fix a point ( y (cid:48)(cid:48) , y n − ) such that y n − > Lθ | y (cid:48)(cid:48) | α .By (2.4) we have ∂φ∂y n − ( y (cid:48)(cid:48) , y n − ) ≤ ∂φ∂y n − (0 , y n − ) + L | y (cid:48)(cid:48) | α . On the other hand, the monotonicity formula (2.5) and the assumption 0 (cid:48) ∈ ∂S (cid:48) u imply ∂φ∂y n − (0 , y n − ) ≤ ∂φ∂y n − (0 , − θy n − = u n − θy n − . By combining the previous two inequalities, it follows that ∂φ∂y n − ( y (cid:48)(cid:48) , y n − ) ≤ u n − θy n − + L | y (cid:48)(cid:48) | α < u n , which means ( y (cid:48)(cid:48) , y n − ) ∈ S (cid:48) u or equivalently y n − > γ ( y (cid:48)(cid:48) ). Thus, the epigraph of the cusp y n − = Lθ | y (cid:48)(cid:48) | α , i.e. (cid:110) ( y (cid:48)(cid:48) , y n − ) : y n − > Lθ | y (cid:48)(cid:48) | α (cid:111) , touches the graph of γ from above. (cid:3) Remark . Note that the opening of the cusp in the proof of Theorem 2.3 is determinedby Lθ . Remark . If Λ is a uniformly convex domain with a C k +1 , k ≥
1, smooth boundarythen it is not difficult to see that ∂S (cid:48) u is a C k graph. Indeed, it suffices to show that γ is a C k function. Since Λ is uniformly convex, φ is uniformly concave or D φ ≤ − θ , for some θ >
0. In particular, ∂ φ∂y n − ≤ − θ <
0. Since γ ( y (cid:48)(cid:48) ) is the only solution of the equation ∂φ∂y n − ( y (cid:48)(cid:48) , y n − ) = 0, by the implicit function theorem, γ is as regular as ∂φ∂y n − , i.e. C k (note that in the smooth case, we may assume without loss of generality that u n = 0 in(2.3)).2.1. Theorem 2.4 is sharp.
In Theorem 2.4, it was shown that for a strictly convexset, the boundary of the shadow is locally a continuous graph in any given direction. Itis natural to wonder if this result extends to merely convex sets. Indeed, the followingcounterexample shows that this is not so: in R , consider the circle { ( x, y, z ) : ( x − + z = 1 } and construct a cone-like set by connecting this circle to the point (0 , ,
0) with linesegments. It is not difficult to see that this process generates a convex body so that whenit is illuminated in the direction (0 , , { ( x, y, z ) : ( x − + z = 1 } ∪ { (0 , t,
0) : 0 ≤ t ≤ } . In particular, the boundary of the shadow is not a graph near the origin in any coordinatesystem.2.2.
Theorem 2.5 is sharp.
Here, we show that Theorem 2.5 is optimal in the followingsense: given a direction u ∈ S n − , there exists a smooth (i.e. C ∞ ) convex body Λ ⊂ R n for which the boundary of the shadow is not locally H¨older continuous; therefore, theuniform convexity assumption is necessary for the conclusion of the theorem. Indeed, thekey observation in the construction of the counterexample is that for a smooth, strictly convex set, γ ( y (cid:48)(cid:48) ) is the unique solution of the equation ∂φ∂y n − ( y (cid:48)(cid:48) , y n − ) = 0 (see (2.3)),and since we are working locally, it suffices to find a smooth, strictly convex function φ : R → R whose level set { ( x, y ) : ∂ y φ ( x, y ) = 0 } is far from smooth. In fact, an example like this already appeared in work of Kiselman[17] in which the regularity of the projection of a three dimensional convex set onto a2-dimensional plane is analyzed (see [17, Example 3.2]): let q be an odd natural numberand set φ ( x, y ) = x (4 − y + 12 y ) + 1 q + 1 y q +1 − q + 2 y q +2 ;note that φ is convex in the strip | y | < and ∂ y φ ( x, y ) = ( y q − x )(1 − y ) . Thus, one may construct a smooth convex set in Λ ⊂ R whose boundary is locally givenby φ in a neighborhood of the origin. In particular, at the local level { ( x, y ) : ∂ y φ ( x, y ) = 0 } is represented by (cid:110) ( x, y ) : y = | x | q (cid:111) , and by selecting u = (0 , ,
0) it becomes evident that illuminating Λ in the direction u generates a shadow boundary which is of class C q . Since q can be taken arbitrarily large,this family of examples shows that for each α ∈ (0 , α so that the boundary of the shadow is not C ,α . Note that this level set method alsosuggests a way of constructing shadows on the surface of convex bodies with a specifieddegree of regularity.3. Regularity of shadows generated by convex projections
Let Ω ⊂ R n , Λ ⊂ R n be two convex domains and suppose that we wish to orthogonallyproject Ω onto Λ. This operation generates a shadow region P Λ (Ω) ∩ ∂ Λ on the boundaryof Λ. The purpose of this section is to study the regularity of this shadow. In other words,given z ∈ ∂ ( P Λ (Ω) ∩ ∂ Λ), we wish to understand how smooth z ∈ ∂ ( P Λ (Ω) ∩ ∂ Λ) is ina neighborhood of z .3.1. Weak case.Theorem 3.1.
Let Ω ⊂ R n be a bounded strictly convex domain and Λ ⊂ R n a convexdomain whose boundary is C , . If Ω ∩ Λ = ∅ , then ∂P Λ (Ω) is finitely ( n − -rectifiable.Proof. Consider an arbitrary y ∈ ∂P Λ (Ω), and let φ : T y Λ → R be a local C , concavechart representing ∂ Λ in a neighborhood B r y around y so that ∇ φ ( y ) = 0; by translatingthe coordinate system, if necessary, we may also assume y = 0. Note that the half-line L at the origin in the direction of the normal of Λ at 0 touches Ω tangentially at some point, say, x (since the projection occurs along the normal to Λ and y ∈ ∂P Λ (Ω)). By convexityof Ω, L lives on a tangent space of Ω at x with normal, say ν . Since (cid:104) ν, N Λ (0) (cid:105) = 0, itfollows that ν lives on the tangent space of Λ at 0. Let e n − := ν and { e , . . . , e n − } be abasis for R n − ; set t ∗ y := dist (Ω , Λ) + 2 max { diam (Ω) , diam ( B r y ) } , and Ψ( z (cid:48) ) := Ψ( z (cid:48)(cid:48) , z n − ) = φ ( z (cid:48) ) − t ∗ y | z | . Note that Ψ is C , and uniformly concave, so by Theorem 2.5, it follows that locallyaround the origin, the level set { ( z (cid:48)(cid:48) , z n − ) : 0 = ∂ z n − Ψ( z (cid:48)(cid:48) , z n − ) } , is a Lipschitz graph which will be denoted by ˜ γ ( z (cid:48)(cid:48) ) = z n − ( z (cid:48)(cid:48) ) (see (2.3)). Now let γ ( z (cid:48)(cid:48) ) := max { ˜ γ ( z (cid:48)(cid:48) ) , } , and note that γ is Lipschitz. We claim that locally around theorgin,(3.1) Φ − ( P Λ (Ω)) ⊂ { ( z (cid:48)(cid:48) , z n − ) : z n − ≤ γ ( z (cid:48)(cid:48) ) } , where Φ( w ) := ( w (cid:48) , φ ( w (cid:48) )). Indeed, let z := ( z (cid:48)(cid:48) , z n − ) ∈ Φ − ( P Λ (Ω)) \ { } ;if z n − ≤
0, then since γ ≥
0, the result follows. So without loss of generality assume z n − >
0. Since z ∈ Φ − ( P Λ (Ω)), it follows that Φ( z ) + t ( z ) N Λ ( z ) ∈ ∂ Ω where t ( z ) > P R n − (Φ( z ) + t ( z ) N Λ ( z )) ⊂ R n − and note that the e n − component of this point is negative (since Ω is strictly convex and e n − is one of its outer normal vectors). In other words, z n − − t ( z ) ∂ z n − φ ( z (cid:48) ) <
0. Thus, ∂ z n − φ ( z (cid:48) ) > z n − >
0) and since t ( z ) ≤ t ∗ y , it follows that(3.2) ∂ z n − Ψ( z (cid:48)(cid:48) , ˜ γ ( z (cid:48)(cid:48) )) = 0 < ∂ z n − φ ( z (cid:48) ) − t ∗ y z n − = ∂ z n − Ψ( z (cid:48)(cid:48) , z n − );now assume by contradiction that z n − > γ ( z (cid:48)(cid:48) ). In particular, z n − > ˜ γ ( z (cid:48)(cid:48) ) so bymonotonicity, (cid:104)∇ Ψ( z (cid:48)(cid:48) , z n − ) − ∇ Ψ( z (cid:48)(cid:48) , ˜ γ ( z (cid:48)(cid:48) )) , (0 , z n − − ˜ γ ( z (cid:48)(cid:48) )) (cid:105) ≤ . Thus, ∂ z n − Ψ( z (cid:48)(cid:48) , z n − ) ≤ ∂ z n − Ψ( z (cid:48)(cid:48) , ˜ γ ( z (cid:48)(cid:48) )) = 0 , and this contradicts (3.2) and proves the claim (i.e. (3.1)). Next, note that0 ∈ Φ − ( P Λ (Ω)) ∩ { ( z (cid:48)(cid:48) , z n − ) : z n − ≤ γ ( z (cid:48)(cid:48) ) } , and since γ is Lipschitz, (3.1) implies that we can place a cone oriented in the direction e n − so that it lies in R n − \ Φ − ( P Λ (Ω)) . The opening of the cone depends on the Lipschitz constant of ∇ Ψ and the uniform con-vexity constant of − Ψ; in particular, it depends on t ∗ y . However, since P Λ (Ω) is bounded(recall that Ω is bounded) and the domains have disjoint closures, it follows that t ∗ y hasa uniform positive lower bound. The existence of this cone implies the claim within theproof of [16, Proposition 4.1]. Indeed, this is the only part where Indrei used the uniformconvexity of Λ, which we were able to replace with strict convexity of Ω in our proof above;thus, the rest of the proof follows exactly as [16, Proposition 4.1] (the idea is that oncewe have a cone at a point, we can use the C , regularity to transition between charts toget a cone at every point of ∂P Λ (Ω); nevertheless, the cones may be oriented in differentdirections, but this readily implies rectifiability via a covering argument). (cid:3) Remark . If in Theorem 3.1 Λ is bounded, then one may replace t ∗ y with t ∗ := dist (Ω , Λ) + 2 max { diam (Ω) , diam (Λ) } . Remark . The disjointness assumption in Theorem 3.1 is necessary: indeed, considera Cantor set C on [1 ,
2] and let g be a smooth function whose zero level set is C . For (cid:15) > f ( x ) := x + (cid:15)g ( x ) is convex, so its epigraph is a convex set in R . Moreover, consider the epigraph of the function h ( x ) := x ; of course, it is likewiseconvex. Now it is not difficult to see that using these epigraphs, one may obtain twobounded convex sets, say Ω and Λ, with the property that their boundaries intersect onthe image of C under h . In this case, ∂ ( P Λ (Ω) ∩ ∂ Λ) does not have finite H measure.Nevertheless, in the general case one may still prove a local version of Theorem 3.1 awayfrom ∂ ( ∂ (Ω ∩ Λ) ∩ ∂ Λ).3.2.
Smooth case.
In Theorem 3.1, we utilized a geometric method of investigating theregularity of shadow boundaries generated by orthogonal projections. In what follows, wedevelop a more functional approach to attack this problem. The idea is to represent theunknown boundary as the level set of a function defined in terms of local charts. However,since the differential of this function contains the information regarding the regularity ofthe level set, we need to ensure that this function is smooth enough; this leads us toimpose higher regularity on the domains.
Theorem 3.4.
Suppose Ω ⊂ R n and Λ ⊂ R n are C k +1 , k ≥ , convex domains separatedby a hyperplane with Ω bounded and uniformly convex. If Ω ∩ Λ = ∅ , then ∂P Λ (Ω) islocally a C k smooth ( n − -hypersurface.Proof. Given a point y ∈ ∂ Λ let f : R n − → R be the C k +1 concave function whichrepresents ∂ Λ locally around y . Likewise, for x ∈ ∂ Ω let g denote the C k +1 uniformlyconcave function locally representing Ω around x . Set F ( y , . . . , y n ) = y n − f ( y , . . . , y n − ) , G ( x , . . . , x n ) = x n − g ( x , . . . , x n − )and consider the function φ : R n × R n × R → R n +3 given by φ ( x, y, t ) := ( G ( x ) , F ( y ) , ∇ G ( x ) · ∇ F ( y ) , y + t ∇ F ( y ) − x ) . Geometric considerations imply that locally ∂P Λ (Ω) = φ − (0 , , , { F ( y ) = 0 } locally describes the boundary of Λ and { G ( x ) = 0 } that of Ω; if ∇ G ( x ) · ∇ F ( y ) = 0, thenthe normal of Λ at y is orthogonal to the normal of Ω at x , and this implies that y = P Λ ( x )is a boundary point of P Λ (Ω); note that in this case, t = t ( x, y ) = | x − y | / |∇ F ( y ) | andthe positive separation implies t >
0. Our goal is to investigate the differential of thismap in order to apply the implicit function theorem. With this in mind, let φ ( x, y, t ) := G ( x ) φ ( x, y, t ) := F ( y ) φ ( x, y, t ) := ∇ G ( x ) · ∇ F ( y )Φ( x, y, t ) := [ φ ( x, y, t ) , . . . , φ n +3 ( x, y, t )] T := y + t ∇ F ( y ) − x. Thus, ∇ x φ = ∇ G ( x ) ∇ y φ = 0 ∂ t φ = 0 ∇ x φ = 0 ∇ y φ = ∇ F ( y ) ∂ t φ = 0 ∇ x φ = D G ( x ) ∇ F ( y ) ∇ y φ = D F ( y ) ∇ G ( x ) ∂ t φ = 0 D x Φ = − Id ∈ R n × n D y Φ = Id + tD F ( y ) ∈ R n × n D t Φ = ∇ F ( y ) . Therefore, D Φ( x, y, t ) = ∇ G ( x ) T ∇ F ( y ) T D G ( x ) ∇ F ( y )) T ( D F ( y ) ∇ G ( x )) T − Id Id + tD F ( y ) ∇ F ( y ) (note that this is an ( n + 3) × (2 n + 1) matrix). The strategy now is to prove ker( D Φ) T = { } at points ( x, y, t ) ∈ φ − (0 , , , α , α , α , v ) ∈ ker( D Φ) T , and note that since D Φ( x, y, t ) T = ∇ G ( x ) 0 D G ( x ) ∇ F ( y ) − Id ∇ F ( y ) D F ( y ) ∇ G ( x ) Id + tD F ( y )0 0 0 ∇ F T ( y ) , we have 0 = α ∇ G ( x ) + α D G ( x ) ∇ F ( y ) − v ;(3.3) 0 = α ∇ F ( y ) + α D F ( y ) ∇ G ( x ) + v + tD F ( y ) v ;(3.4) 0 = ∇ F ( y ) · v. (3.5)In particular, 0 = ∇ F ( y ) · v = ∇ F ( y ) · ( α ∇ G ( x ) + α D G ( x ) ∇ F ( y ))= α ∇ F ( y ) · ∇ G ( x ) + α ∇ F T ( y ) D G ( x ) ∇ F ( y )= α ∇ F T ( y ) D G ( x ) ∇ F ( y ) , (note ∇ F ( y ) · ∇ G ( x ) = 0 since ( x, y, t ) ∈ φ − (0 , , , G is uniformly convex, itfollows that α = 0 and so (3.3) implies v = α ∇ G ( x );plugging this information into (3.4) and taking a dot product with ∇ G ( x ) yields0 = α (cid:0) t ∇ G ( x ) T D F ( y ) ∇ G ( x ) + |∇ G ( x ) | (cid:1) . Since |∇ G ( x ) | >
0, and F is convex, it follows that α = 0 which readily implies v = 0 andso α = 0. Thus, we proved ker( D Φ T ) = { } ; in particular, rank ( D Φ) = n + 3 for eachpoint of interest ( x, y, t ). We may now use the implicit function theorem to conclude. (cid:3) Remark . The disjointness assumption in Theorem 3.4 is necessary, cf. Remark 3.3.4.
The singular set associated to a Monge-Amp`ere equation
In this section, a connection is established between the illumination shadow, the projec-tion shadow, and the singular set associated to a Monge-Amp`ere equation arising in masstransfer theory. More precisely, we apply the results of the previous sections to improvea result of Indrei [16] (see § The structure of the singular set.
In order to analyze the singular set for thefree boundaries, we recall two sets which play a crucial role in the subsequent analysis;cf. [16, Equations (2.2) and (2.3)]. The nonconvex part of the free boundary ∂U m ∩ Ω isthe closed set(4.1) ∂ nc U m := { x ∈ Ω ∩ U m : Ω ∩ U m fails to be locally convex at x } . Moreover, the nontransverse intersection points are defined by(4.2) ∂ nt Ω := { x ∈ ∂ Ω ∩ Ω ∩ ∂U m : (cid:104)∇ Ψ m ( x ) − x, z − x (cid:105) ≤ ∀ z ∈ Ω } , where ˜Ψ m is the extension of Ψ m given by [9, Theorem 4.10]. By duality, ∂ nc V m and ∂ nt Λare similarly defined. Now, for x ∈ ∂ (Ω ∩ U m ) let L ( x ) := (cid:110) ∇ ˜Ψ m ( x ) + x −∇ ˜Ψ m ( x ) | x −∇ ˜Ψ m ( x ) | t : t ≥ (cid:111) ; K := (cid:110) x ∈ ∂ (Ω ∩ U m ) : L ( x ) ∩ Ω ∩ U m ⊂ ∂ (Ω ∩ U m ) (cid:111) ; S := ∇ ˜Ψ − m ( ∂ nt Λ) ∩ K ; A := S ∩ ∂U m ; A := S \ ∂U m . The singular set of the free boundary ∂V m ∩ Λ is S = ( ∇ ˜Ψ m ( ∂ nc U m ) ∪ ∇ ˜Ψ m ( S )) ∩ ∂V m ∩ ∂ Λ= ( ∇ ˜Ψ m ( ∂ nc U m ) ∩ ∂V m ∩ ∂ Λ) ∪ ( ∇ ˜Ψ m ( S ) ∩ ∂V m ∩ ∂ Λ)= ( ∇ ˜Ψ m ( ∂ nc U m ) ∩ ∂V m ∩ ∂ Λ) ∪ ∇ ˜Ψ m ( A ) ∪ ∇ ˜Ψ m ( A ) , see [16, Theorem 4.9]. The next lemma describes the first two sets appearing in S . Lemma 4.1.
Assume Ω ⊂ R n and Λ ⊂ R n are strictly convex bounded domains withdisjoint closures. Then ( ∇ ˜Ψ m ( ∂ nc U m ) ∩ ∂V m ∩ ∂ Λ) ∪ ∇ ˜Ψ m ( A ) is H n − σ -finite. Moreover, if Ω is C , then H n − (( ∇ ˜Ψ m ( ∂ nc U m ) ∩ ∂V m ∩ ∂ Λ) ∪ ∇ ˜Ψ m ( A )) < ∞ . Proof.
For y ∈ ( ∇ ˜Ψ m ( ∂ nc U m ) ∩ ∂V m ∩ ∂ Λ) set x := ∇ ˜Ψ ∗ m ( y ); since Ω is convex and x ∈ ∂ nc U m , it follows that x / ∈ ∂ Ω \ ∂U m . Moreover, since free boundary never maps tofree boundary (see e.g. [16, Proposition 2.15]), we also have x / ∈ ∂U m ∩ Ω, which implies x ∈ ∂U m ∩ ∂ Ω. Therefore,( ∇ ˜Ψ m ( ∂ nc U m ) ∩ ∂V m ∩ ∂ Λ) ⊂ ∇ ˜Ψ m ( ∂U m ∩ ∂ Ω) ∩ ∂V m ∩ ∂ Λ . An application of [16, Proposition 4.8] yields that ∇ ˜Ψ m ( ∂ nc U m ) ∩ ∂V m ∩ ∂ Λis H n − - finite; the fact that ∇ ˜Ψ m ( A ) is H n − σ -finite ( H n − finite if Ω is C ) followsfrom [16, Corollary 4.6]. (cid:3) In the following lemma, we establish a connection between the singular set S and theboundary of the projection of Ω onto Λ studied in § Lemma 4.2.
Assume Ω ⊂ R n and Λ ⊂ R n are strictly convex bounded domains withdisjoint closures. Then (4.3) ∇ ˜Ψ m ( A ) ⊂ ∂P Λ (Ω) . Proof.
Let y := ∇ ˜Ψ m ( x ) ∈ ∇ ˜Ψ m ( A ), L t := ∇ ˜Ψ m ( x ) + x −∇ ˜Ψ m ( x ) | x −∇ ˜Ψ m ( x ) | t and note that thehalf-line { L t } t ≥ is tangent to the active region. Since x ∈ ∂ Ω \ ∂U m , it follows that L t istangent to Ω at x ; hence, it is on a tangent space to Ω at x . Let z = P Λ ( x ) ∈ ∂ Λ (recallthat P Λ is the orthogonal projection operator). Then by the properties of the projection(and the convexity of Λ), x − z is parallel to some normal N Λ ( z ) of Λ at z . Since x ∈ S ,it follows that ∇ ˜Ψ m ( x ) ∈ ∂ nt Λ; in particular, x − ∇ ˜Ψ m ( x ) is parallel to N Λ ( ∇ ˜Ψ m ( x )).Thus, by uniqueness of the projection, it readily follows that z = ∇ ˜Ψ m ( x ) = y . Combining { L t } t ≥ ⊂ T x Ω and y = P Λ ( x ) yields y ∈ ∂P Λ (Ω). (cid:3) Lemmas 4.1 & 4.3 imply that the singular set S is contained in the union of an H n − σ -finite set and ∂P Λ (Ω) under a strict convexity and disjointness assumption on the domains.Thus, a way to obtain bounds on the Hausdorff dimension of the singular set is by studyingthe Hausdorff dimension of ∂P Λ (Ω). In [16, Proposition 4.1], Indrei shows that if Ω isa bounded convex domain and Λ is uniformly convex, bounded, and C , smooth, then P Λ (Ω) ∩ ∂ Λ is ( n − ∂ ( ∂ (Ω ∩ Λ) ∩ ∂ Λ); in particular, if the domainshave disjoint closures, then H n − ( ∂P Λ (Ω)) < ∞ . The proof of [16, Proposition 4.1] is technical but relies on a simple idea which we describein the language developed in this paper in order to further highlight the connection withshadows: let y ∈ ∂P Λ (Ω) and x ∈ ∂ Ω be such that y = P Λ ( x ). Then ∂P Λ (Ω) ⊂ ∂ Λ \ S N Ω ( x ) ,where N Ω ( x ) is any normal of Λ at the point x and S N Ω ( x ) is the shadow from §
2. In otherwords, P Λ (Ω) is trapped in the illuminated portion of ∂ Λ under parallel illumination in the direction N Ω ( x ). Since y ∈ ∂P Λ (Ω) ∩ ∂S N Ω ( x ) , it follows that ∂S N Ω ( x ) acts as aone-sided support for ∂P Λ (Ω) locally around y . Therefore, if one can place a cone inthe shadow portion S N Ω ( x ) , a compactness argument would yield the desired rectifiabilityresult. Indeed, this is where the uniform convexity and C , assumptions come intoplay in the proof of [16, Proposition 4.1]. However, the results of § C , regularity assumption is irrelevant: indeed, § C ∞ strictly convex set whose shadow is H¨older, but not Lipschitz (with arbitrarilysmall H¨older exponent). In particular, this shows that one may not hope to remove theuniform convexity assumption by the same method (i.e. by using ∂S N Ω ( x ) as a supportfunction). However, Theorem 3.1 implies that one one may obtain the cone without auniform convexity assumption; this is achieved by cooking up a new type of supportfunction related to the distance between the two sets. Moreover, Theorem 3.4 yields ahigher regularity result. With this discussion in mind, we obtain the following theoremwhich improves [16, Theorem 4.9]: Theorem 4.3.
Assume Ω ⊂ R n , Λ ⊂ R n are bounded strictly convex domains and that Λ has a C , boundary. If Ω ∩ Λ = ∅ , then the free boundary ∂V m ∩ Λ is a C ,αloc hypersurfaceaway from the compact, H n − σ -finite set: S := ( ∇ ˜Ψ m ( ∂ nc U m ) ∩ ∂V m ∩ ∂ Λ) ∪ ∇ ˜Ψ m ( A ) ∪ ∇ ˜Ψ m ( A ) . If Ω has a C boundary, then S is H n − finite. Moreover, if Ω and Λ are C k +1 , k ≥ ,and Ω is uniformly convex, then ∇ ˜Ψ m ( A ) is contained on an ( n − -dimensional C kloc hypersurface.Remark . By duality and symmetry, an analogous statement holds for ∂U m ∩ Ω. Remark . One may remove the disjointness assumption and obtain corresponding re-sults by utilizing the method in [16, § Acknowledgments.
This work was completed while the first author was a Huneke Post-doctoral Scholar at the Mathematical Sciences Research Institute in Berkeley, Californiaduring the 2013 program “Optimal Transport: Geometry and Dynamics,” and while thesecond author was a Postdoctoral Fellow at the Instituto Superior T´ecnico. The excellentresearch environment provided by the University of Texas at Austin, Australian NationalUniversity, MSRI, and Instituto Superior T´ecnico is kindly acknowledged.
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Proc. Amer. Math. Soc. , 103 (1988) 586-590. Emanuel IndreiMSRI17 Gauss WayBerkeley, CA 94720email: [email protected]