on the regularity of the solution of the n-dimensional ...js/math646/cheng-yau.minkowski.pdf496...

COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, VOL. XXIX (1976)

On the Regularity of the Solution of the n-Dimensional Minkowski Problem

SHIU-YUEN CHENG AND SHING-TUNG YAU* Princeton University Stanford University

0. Introduction

Given a closed strictly convex hypersurface M in the Euclidean space I?"+*, the Gauss map of M defines a homeomorphism between M and the unit sphere S". Therefore the Gauss-Kronecker curvature of M can be transplanted via the Gauss map to a function defined on S". If this function is denoted by K, then Minkowski observed that K must satisfy the equation

where xi are the coordinate functions on S". Minkowski then asked the converse of the problem. Namely, given a

positive function K defined on S" satisfying the above integral conditions, can we find a closed strictly convex hypersurface whose curvature function is given by K? Minkowski solved the problem in the category of polyhedrons. Then A. D. Alexandrov and others solved the problem in general. However, this last solution does not provide any information about the regularity of the (unique) convex hypersurface even if we assume K is smooth.

In the two-dimensional case, H. Lewy was the first one who proved that if K is analytic, the solution to the Minkowski problem is also analytic. Around 1953, A. V. Pogorelov [9] and L. Nirenberg [6] solved the regularity problem in the smooth category independently. Their methods were quite different and restricted only to two dimensions. The method of Pogorelov was to show that the (unique) generalized solution of Alexandrov is smooth. This depends on the solvability and regularity of the Dirichlet problem of the two-dimensional Monge-Ampkre equation. The method of L. Nirenberg was to use the continuity method to produce a smooth solution directly. This depends on a priori estimates which are available only for elliptic equations of two variables.

*The research for this paper was carried out at the Courant Institute of Mathematical Sciences under the National Science Foundation Grant MPS-73-08487-A02 and supported by a Sloan Fellowship. Reproduction in whole or in part is permitted for any purpose of the United States Government.

495 0 1976 by John Wiley & Sons, Inc.

496 SHIU-YUEN CHENG AND SHING-TUNG YAU

For dimensions higher than two, the regularity problem was unknown until A. V. Pogorelov announced the solution in [8]. He derived in this paper an estimate of the derivatives of third order using a computation of E. Calabi. Then, without further proof, he claimed the rest of the proof is the same as the one in two dimensions. However, as was indicated above, his proof in two dimensions depends on the regularity of the Monge-Ampbre equation. In higher dimensions, this was not known at that time. Two years later, he announced in [7] the regularity of the higher-dimensional Monge-Amp‘ere equation. Unfortunately, besides the gaps in his proof (see the comments in [4]) the proof of the regularity depends on the solution of the regularity of the Minkowski problem.

The purpose of this paper is to provide a proof of the regularity of the higher-dimensional Minkowski problem without assuming the regularity of the Monge-Ampkre equation. Our method is the continuity method which was used by L. Nirenberg in [6]. However one notices that even if we assume the a priori estimates of the third derivatives of the Monge-Ampkre equation, Nirenberg’s method cannot be generalized immediately. For example, Bon- net’s theorem tells us that for a complete Riemannian manifold with Ricci curvature bounded from below by a positive constant, the diameter can be estimated from above depending only on that constant. This theorem is readily applicable to the two-dimensional Minkowski problem but not to the higher-dimensional problem. Furthermore, since we have only interior estimates for the third derivatives, we need Lemma 4 which is not easy. The strategy of studying differential equations on the sphere was also used by L. Nirenberg. But the generalization is not trivial.

For the purpose of proving the regularity of the Monge-Ampbre equation in [4], we briefly recall some known facts about the generalized Minkowski problem. The important fact that we need in [4] is that when a sequence of area functions of convex bodies converges to an area function of a convex body in the I,’-sense, then a subsequence of the support functions of the convex bodies converge uniformly to the support function of the corresponding convex body.

Finally we would like to thank Professors Calabi, Chern, Kohn, Moser and Nirenberg for their encouragement in the writing of this paper.

1. A Priori Estimates In this section, we record the estimates obtained by Pogorelov [7], [8]

where,. in the case of the third-order estimates, an essential idea of Calabi [3] was used.

Let CR be a n n-dimensional convex domain a n d FE C’(fi) with F>O. Let U E C4(R)n CoS1(ft) be a strictly convex solution of the equation det(u, , )=F(x) on SL a n d assume that ulan= a. Then there is a constant Q!

LEMMA 1.

THE SOLUTION OF THE n-DIMENSIONAL MINKOWSKI PROBLEM 497

depending only on n, maxdL IVu(, maxn Iu - a1 and the C1.'-norm of F on fi such that max, ~ u , , ( x ) ~ S a 1u - a/-' .

Proof: For any fixed direction alax,, Pogorelov [7] considered the function w = ( a - u)e"32u,1 on the domain R. Then m e may assume that w attains its maximum at some point in a. (If u,, were not bounded in fi, we could consider the function ( a - E - u)eu;zu,, in the domain {x I u(x) 5 a - E } and let

With a tricky calculation, Pogorelov derived the following inequality: E -3 0.)

nuii Uf

a - u ( a -u )2 _ _ _ _ ~ + u % + u ~ u ~ ~ F - ~ F ~ + F - ~ F ~ ~ - F ~ F : s o .

Multiplying (1.1) by ( a - u)' e "', one arrives at the inequality

(1 2) W ~ + A W + B S O ,

where A and B admit estimates depending on the quantities mentioned in the lemma.

The conclusion of the lemma follows readily from (1.2). If the function F(x) is bounded away from zero, then it is clear that an

upper estimate of the quantities u,, also provides a lower estimate of u,,. Hence if we assume that F is positive in R, then for a precompact subdomain of R we have upper and lower estimates of u , ~ .

In order to have an interior estimate of the third derivative of u, it suffices therefore to estimate the quantity C,,J,k,o,P,y uLa U J P U k y U l J k U a P y . If one gives the graph of u the affine metric defined in [4], the last quantity is dominated by the affine length of the Fubini-Pick form. In an intrinsic manner, this length was estimated in [4]. However, previous to [4], Pogorelov had already obtained the third-order estimates in a somewhat different manner. The essential ideas of all these approaches were due to E. Calabi [3].

LEMMA 2. Let R be a n n-dimensional convex domain a n d FE C3(fi) with F>O. Let u E C5((n) fl C'3'(fi) be a strictly convex solution to the equation det (14,~) = F(x) and assume that uIan = a. Then the third derivative of u can be estimated by the C1"-norm of u, the C2,'-norm of F, maxXEn 1/F(x) and the distance from an.

2. The Setting of the n-Dimensional Minkowski Problem Let K be a positive Ck3"-function defined on the unit sphere S". Here

k 2 3 and 1 > a > 0. Suppose, for all coordinate functions x, on S", we have


Then the Minkowski problem is to find a convex hypersurface M in Rn+' such that under the identification of the Gauss map, the Gauss-Kronecker curvature of M is given by K.

Given a strictly convex hypersurface M, we can define its support function H as follows: Let x = ( x l , - * - , x,+~) E S" be any unit vector. Then we can find a (unique) point y = ( y l , - , yn+J E M so that the Gauss image of y is x. By definition we have H ( x ) =I x i y i . Extending this definition to R"+'\{O}, we obtain a homogeneous function of degree one by the equation H ( x ) =

By the strict convexity of M, the coordinates y l , * * * , ynil of M may be considered as functions of x l , - . , xntl with CY:'=': x : = 1. We extend these functions to be defined for all values of (xl, * * , x , + J by requiring them to be homogeneous of degree zero. From the definition of H, it follows that

1x1 H ( x / I x I ) .

We can therefore recover the coordinate functions of M by the equations

(2.3) C3H

Yi 'aX, for i = l ; - - , n + l .

determined by their values on the hyperplanes xi = -1 for i = 1, - * * , n + 1.

can also be defined by

Since y i are functions of homogeneity of degree zero, they are completely

As a typical example, we shall consider the hyperplane x,+~ = -1. Since H

H is clearly a convex function. We shall restrict H to the hyperplane ~ , , + ~ = - 1 and compute its Hessian.

First of all, we note the following general fact. Let {e l , . - - , en} be an orthonormal frame of the sphere S". Then restricting H to S" and taking its Hessian ( I f i j ) with respect to this frame, we have

( 1 + 2 det (*)(XI, axi axj . . - , x , , -1)

i = l

(2.4) X n -1

= det (Hi j + H a i i )

THE SOLUTION OF THE n -DIMENSIONAL MINKOWSKI PROBLEM 499

Since (2.4) is a straightforward computation, using the homogeneity of H, we omit its calculation. On the other hand, it is a familiar fact (cf. [6]) that the right-hand side of (2.4) is given by 1/K, where K is the Gauss-Kronecker curvature of M considered as a function on S".

Hence extending K to be a function of homogeneity of degree zero, we have

a2 H i = l axi axj

(2.5) (1 + 2 x;),''+' det (-)(.it A . - , xnr -1) = (K(x,, - . - , x,, -1))-' .

Obviously, equations similar to (2.5) should hold over the other hyperplanes xi = -1, i 5 n. All these equations can combine into the following one on the unit sphere:

1 K

det (Hij + H aij) = - .

It follows easily from (2.5) that H is strictly convex on the hyperplanes x, = -1.

Conversely, suppose we can find a convex function of homogeneity of degree one satisfying equations similar to ,(2.5) (or equivalently (2.6)), then we can find a strictly convex hypersurface whose support function is H and whose curvature function is K. Explicitly the coordinate functions of this hypersurface are obtained by the formulae (2.3) so that the smoothness of the hypersurface follows from the smoothness of H.

To see this assertion, we note that (2.3) defines functions with homogeneity of degree zero so that it is sufficient to consider their effects after restricting them to the hyperplanes xi = -1. For simplicity we consider the strictly convex function u(xl, - * . , x,) = H(xl, . . . , x,,, -1).

Let a* ={(ul(x), * * * , u, (x ) ) I x E R"}. Then the Legendre transform of u is the strictly convex function u* defined on a* by the equation

n

U * ( U l ( X ) , * * , u,(x)) = c xiui(x)- u(x) . i = l

(2.7)

The normal of the graph of u* is given by

= (1 + t x:)1'2(x1, * - * , x,, -1). i=l


The support function of the graph of the function u* at the point (1 +Cy=l xf)-1/2(x1 , - * , x,, -1) is therefore equal to

Therefore, its value at (xl, . . * , x,, -1) is exactly H(xl, . . . , xn, -1). It remains to observe that

This follows immediately from the homogeneity of degree one of H. In conclusion, the problem of Minkowski is reduced to the solution of the

equation (2.6) on the unit sphere S" under the condition that K is positive and satisfies the equation (2.1).

To solve this problem, we use the continuity method (cf. [6 ] ) . Thus let

for 0 S t S 1. Then clearly all the functions K, are positive and satisfy the equation (2.1).

Let S, = { t ~ [0, 11 1 the equation det (Hz , +Ha,,) = 1/K, has a C5@-solution H on the sphere S" such that (H,+Ha,,)>O}. The method of continuity is divided into two steps. The first is to prove that S, is closed in [0, 11 and the second is to prove that S, is open in [0,1]. Since O E S,, this clearly provides S , = [0,1] and the equation (2.6) is solvable. The first step will be carried out in Section 3 and the second step will be carried out in Section 4.

3. The Closedness of the Set S ,

Let { t l } be a sequence in S which converges to a point t o E I O , 11. Let H' be a sequence of convex functions of homogeneity of degree one so that, restricting to S", H' are Ckt2@-functions with (Hfj+ H' 6,) > 0 and satisfying the equations det (H!j+ H' tiii) = l lKi, .

From the discussions in Section 2, we know that equations y: = aH'/axi define convex hypersurfaces M' whose support functions are given by H' , respectively. We claim that the outer diameters of M' are uniformly bounded (independent of 1).

Since we shall need it in the next paper, we state a lemma in a more precise form.

THE SOLUTION OF THE &DIMENSIONAL MINKOWSKI PROBLEM 501

LEMMA 3. Let M be a compact convex C4-hypersurface in R"+'. Let K be its Gauss-Kronecker curvature function defined on Sn. Then the extrirtsic diameter L of M can be estimated from above by the quantity

where c, is a positive constant depending only on n.

Proof: The lemma is known in the nonsmooth case (see [2]). (The assumption that M is a C4-hypersurface is made to insure that K is defined. Of course we can replace this by the concept of generalized curvature.)

Let L be the extrinsic diameter of M. Then there are two points {p, q} on iv2 such that the line segment joining p and q has length L. Without loss of generality, we may assign our origin to be the midpoint of this segment. Let u be a unit vector in the direction of this segment. Then for any unit vector w E S", the support function of M at w is given by H(w) = S U P , , ~ ~ ( Y , w) 2 11. max (0, (u, w)).

Integrating over the sphere S", we have

(3.1)

The volume element of M is 1/K times the volume element of S". Hence

(3.2)

To compute H, we note that in R"+l I, (3.3)

Applying the divergence formula to (3.3), it yields

1 Vol (a) = - (3.4) n + l I , H

where I\;i is the volume of the body bounded by M.

The surface element of M is given by 1/K. Hence the lemma is a 6" consequence of (3.1), (3.2), (3.4) and the isoperimetric inequality.


It follows easily from Lemma 3 that the outer diameters of the convex hypersurfaces M' are uniformly bounded from above. We are going to prove that the inner diameters of M' are also bounded from below by a positive constant.

LEMMA 4. In Lemma 3, we can find always a positive constant r which

depends only on an upper estimate of 1/K and a lower estimate of

infuEsn max (0, ( u , w))K(w)- ' so that we can always put a ball of radius r

inside the hypersurface M.

Proof: First of all, we find a lower bound of the areas of the projections of the surfaces on all possible hyperplanes.

Suppose we project the hypersurface M along a unit vector u onto a hyperplane perpendicular to u. Then clearly the volume of the body bounded by M is not greater than the area of the projection multiplied by the outer diameter of M. By (3.4), (3.2) and (3.1), we conclude that the area of the projection is not less than

I,. 1.

Since the volume of the body bounded by M is dominated by L", (3.1) gives us a lower estimate of L. Hence we can find two points p1 and p 2 on M whose distance can be estimated from below. Consider the projection of M on a hyperplane containing the segment p1p2. Since the outer diameter is bounded and the projected area is bounded from below, we can find a point p3 on this hyperplane such that all the angles of the triangle p 1 ~ 2 p 3 lie between E and T - E , where E is a positive constant estimated from below. Lifting p3 to a point p3 in M , we obtain a triangle ~ 1 ~ 2 ~ 3 with similar properties.

If n > 2 , then we project M into a hyperplane containing the triangle ~ 1 ~ 2 ~ 3 . Arguing as above, we can find a point p4 in M such that the angles between all line segments issuing from p4 to the p i , 1 S i S 3 , and the plane of the triangle p, ~ 2 ~ 3 lie between E and T - E , where E is a positive constant estimated from below. Continuing in this way, we obtain n points p l , - * - , pn in M such that the angles between all line segments issuing from pi to the p i , 1 5 j 5 i - 1, in the plane of the simplex plp2 * * ' p,-l lie between E and T - E , where E is a positive constant estimated from below.

Finally, we project the hypersurface M along a direction which is perpendicular to the hyperplane of the simplex p1p2 - 7 . pn. Then arguing as before, we can find a simplex fi1fi2 - fin+' with properties similar to those of


p l p z . . . p , . The simplex P I P 2 . . * Pn+l must lie in the body bounded by M and the properties ensure us of having a ball in its interior whose radius can be estimated from below. This finishes the proof of Lemma 4.

Putting both Lemnld 3 and Lemma 4 together, we see that we can translate the M ' so that the origin lies in the interior of all the M I and the numbers infl infXEMI llxll and supl llxll are positive (finite) numbers. Corresponding to these translations, we change the support functions H' by linear functions so that infl inf,,,. H ' ( x ) > 0 and supl sup,,,. H ' ( x ) <m.

We are now in a position to apply Lemmas 1 and 2 to prove the closedness of S,.

Recall that, on a convex domain 52, any convex function u defined on R admits the estimate

( V u ( x ) ( 5 sup u - u ( x ) d(x , an)-' . ( a 1

It follows from convexity of H', from the fact that supl sup,,,. H ' ( x ) < m, and from the homogeneity of H' that we have a uniform gradient estimate of the H' on each compact set of each hyperplane x i = -1.

Since

(3.5)

one can see easily that for any positive number c, the sets 52,(c)= {(x,, . . . , x,, -1) I H'(xl, * * , x,, -1) < c} lie in a fixed compact set.

For any point (al , - , a,, -l), we can find a positive number c depending only on supl supxEs" H ' ( x ) and Cy=l af such that (al , * * # , a,, -1) E Ol($c) for all 1.

Since all the sets O l ( c ) lie in a fixed compact set, the gradients of all the convex functions H' are uniformly bounded and we can apply Lemmas 1 and 2 to the equations

a2H' (3.6) (1 + 2 x?)~"+' det (-)(xl, axi axj - , x,, -1) = (K,(x l , * . - , x,, -l))-'

i = l

to obtain uniform estimates of the second and the third derivatives of H'. (Note that we clearly have Ck*"-estimates of the functions K'.)

A subsequence of the functions { H ' } therefore converges to a C2*'- function H' which satisfies the following equation:

(3.7) (1+ 2 x ~ ) ~ ' 2 + 1 d e t ( - ) ( x l , . . ~ , x , , - l ) = ( K , ( x l , . . ~ , x , , - 1 ) ) - ' a* H0 i = l axi axj

5 04 SHIU-YUEN CHENG AND SHING-TUNG YAU

Linearizing this equation and using the standard Schauder estimates, one can then prove that H o is a Ck+'@-function and to€ S,. This finishes the proof of the closedness of S,.

4. The Openness of S,

In this case, we shall apply the implicit function theorem to the equation deg ( H i j + H S i j ) = K-'. Here H is considered to be a function defined on the unit sphere.

Let LH be the linearized operator of the operator H + det ( H i j + H S i j ) . Then for all smooth functions u defined on S" we have

where c(Hij + H S i j ) are the cofactors of the matrix (Hi j + H aij). We claim that, when u and 2, are functions in C'(S"),

(4.2)

It is rather easy to see that the equation (4.2) is equivalent to the equation

(4.3) 1 c ( H i j + H 6 i j ) j = 0 i

for all i. (Differentiations here are covariant differentiations on S".) We shall prove (4.3) under the assumption that det (Hij + H a i j ) # 0 in a

neighborhood of the point which we are considering. The general case follows from this by approximatingrH (in a neighborhood of this point).

Clearly, we have

From (4.4), we know that

det ( H p q + H S P q ) 1 C ( H , , + H S , ~ ) ~ i

THE SOLUTION OF THE &DIMENSIONAL MINKOWSKI PROBLEM 505

It remains to show that the terms in the bracket are zero. These are the

Let {oi} be a local orthonormal frame field and {wij} the corresponding commutation formulas and we derive them as follows:

connection forms. Then, by definition,

(4.7)

(4.8)

Exterior differentiating (4.7), we obtain

where Rlikj is the curvature tensor of the sphere. Hence

H. . - H . . = -1 H I R I . . ilk rkl LkJ (4.10) 1

= -Hk 6ij + Hj 8ik .

By exterior differentiating (4.6), one finds that ( H i j ) is symmetric and so

and our assertion (4.2) is proved.

coordinate function x i , that We claim now, for any Css"-function u defined on the sphere and for any

In fact

I-


defines a differentiable functional on the Banach space of C5,"-functions defined on S". The FrechCt derivative of this functional at a function 0 is

given by xiL,(u). Since L,(xi) = 0, (4.2) shows that the FrechCt derivative

of Li is identically zero. Since Li(x i )=O, Li is zero and (4.12) is proved. 6.

Remark. In applications, u is the support function of some strictly convex body and (4.12) can be proved more directly.

We are now in a position to prove the openness of S,. Let B1 be the Banach space of Ck+',"-functions defined on S" and B2 the Banach space of Ck,"-functions f defined on S" such that, for all coordinate functions x i , r

xi f = 0. Then according to (4.12), we can define a transformation F : B1 4 S"

B2 by

(4.13) F( u) = det ( uii + u Sij)

We claim that if H is the support function of some strictly convex hypersurface such that H>O on S", then for functions f in a neighborhood of det (Hij + H a i j ) (in the topology of B2) we can always solve the equation F ( u ) = f , where u is also the support function of some strictly convex hypersurface.

As is well known (cf.'[lO]), it suffices to verify that the linearized operator of F at H is surjective. That is to say, the linear operator LH(u)= Ci,j c(Hij + H Sij)(uij + u Sij) is surjective.

First of all, we find the kernel of LH.

LEMMA 5. Let u be a function in C2(S") such that LH(u)=O, where

This lemma is closely related to the infinitesimal rigidity of the

(Hij + H 6,) > 0. Then, for some constants al , - . , a,+l, u = xYL'=': aixi.

strictly convex body determined by H. Proof:

Consider the vector-valued function 2 = Cy=l uiei + ue,,, , where {e l , - * 9 , en} is a local orthonormal frame field and is the unit outer normal of S" (the subscript of u means the differentiation with respect to the frame {el, , en}). Then u = (Z, entl) and

(4.14)

THE SOLUTION OF THE %DIMENSIONAL MINKOWSKI PROBLEM 507

Let X = C r = l H i e i + H e , + l . Then as in (4.14), we have

(4.15)

Consider the (n - 1)-form a= X A 2 A dZ A d X A * - - A dX, where d X appears (n - 2)-times. By computation,

(4.16) d R = d X A Z A d Z A d X A . . * r \ d X + X r \ d Z / 2 d Z A d X A * . - A d X

The first term on the right-hand side of (4.16) is zero because d Z r \ d X A - * A dX, where dX appears (n - 1)-times, is

Integrating (4.16) over S", we then obtain

Let Zj = Xi (Hii + H Sij)ei. Then

Let

By diagonalizing (Hij + H S i j ) , we may assume that u k j and ujk either have the same sign or are both equal zero. Then by putting (4.18) into (4.17),

(4.19) ( X , en+l)( (uiiujj - uijuji) det (Hij +H6,)-' = 0 . i # j

LH(u) = 0 implies that Ci uii = 0. Therefore,

On the other hand, det (Hi, + H 6 i j ) and ( X , en+l) = H are positive. Therefore,


(4.19) and (4.20) together show that uij = 0 for all i, j . Hence

(4.21) uij + u 6, = 0

for all i, j.

some constants a l , . . . , anil . Putting (4.21) into (4.18), we find that Z=const . and u =C::l' a,x, for

Finally we prove the surjectivity of the linearized operator LH. Let H"(S") be the Sobolev space of functions defined on S" with

derivatives up to order rn. Then LH can be considered as a bounded linear map from H ~ + ' ( s " ) to H ~ ( S " ) .

First of all, choose k so large that C2(S") 2 Hk+'(Sn). This is possible by the Sobolev lemma. With this choice, we can apply Lemma 5 and conclude that the kernel of LH is given by the space of linear functions.

Therefore it follows from (4.2) that the kernel of the adjoint of LH is the space of linear functions. Standard Hilbert space theory shows that, for every function f in H k ( S " ) , we can find a solution U E H ~ + ~ ( S " ) solving the equation LH(u) = f. The standard argument using the interior Schauder estimate also shows that, when f~ Ck3a(Sn), u E Ck+2,a(Sn).

This finishes the proof of the surjectivity of LH and hence the closedness of S , under the assumption that k is large enough to guarantee that C2(S") c Hkt2(Sn). Therefore, for this k, we have solved the equation det (ElLJ + H S , , ) = K-' in the manner that, when K-'E CkZa(Sn) and verifies

If we merely assume that K is a positive function in C3(S") , then we can approximate K in C2z'(S")-norm by positive functions Kl in Ck,a (Sn) . In order to ensure the compatibility condition, we simply replace Kl by El so that

(2.1), H E Ck+'yS").

where

Furthermore xiE;' = 0 and { E l } -+ K in the C2"(S")-norm. By the above

arguments, we can solve the equation det (H , + H Sii) = I?;' with solutions in Ck+29a(Sn) . The arguments in proving the closedness of S, show that we can pick a convergent subsequence of these solutions to converge to a solution of

6.

THE SOLUTION OF THE n-DIMENSIONAL MINKOWSKI PROBLEM 5 09

det ( Hij + H tiij) = K-'. In conclusion, we have the following

THEOREM 1. Let K be a positive function in Ck(S") with k Z 3 . Suppose

x ,K- '= 0 for all coordinate functions xi. Then we can solve the equation

det (Hii + H a , ) = K-', where the solution belongs to Ck+l ,a (Sn) for all 1 > a > O . If K is analytic, then the solution is also analytic. Therefore we can find a compact strictly convex hypersurface in R"+' whose support function is given by H and whose Gauss-Kronecker curvature function is K. Moreover, any two such hypersurfaces must coincide after a translation.

Only the uniqueness part needs a proof. This is a consequence of Theorem 3 in Section 5.

Proof:

5. Generalized Minkowski Problem

In this section, we shall briefly discuss the classical approach to the Minkowski problem.

Let M be a compact convex hypersurface in R"+l. Then the generalized Gauss map G : M + S" is a set-valued map which maps every x E M to the set of all outward normals of the supporting planes of M passing through x. From this map we can define a measure p ( M ) on S" called the area function of M by setting

p ( M , E ) = n-dimensional volume of {x E M : G(x) rl E Z 0) for any Bore1 subset E of S " .

(The complete additivity of p ( M ) follows from the fact that, at almost every point of M , there is a unique supporting plane.)

When. M is strictly convex and smooth, p ( M ) is the absolutely continuous measure on S" represented by 1/K, where K is the Gauss-Kronecker curvature of M transplanted to S" via the Gauss map. When M is a polyhedron, p ( M ) is equal to Ef=, c, 6 , , where 6 , is the unit point mass at ui and c, is the n-dimensional measure of the face of M with outward normal equal to u,.

To study the convergence of compact convex sets, we define a metric A on % = { K ; K is a compact convex set in R"+') in the following way;

A(K, K' ) = sup d(x, K)+sup d(x, K' ) X € K ' x e K

for any K, K'E %.


That A is indeed a metric on % can be easily verified.

PROPOSITION 1. Suppose M, Mi E %, 15 i < m, and M, + M in the topotogy defined by A. Then p ( M i ) + p(M) weakly, i.e., for any f e C(S") , r r

J J

Proof: The proposition is a direct consequence of the following two lemmas.

LEMMA 8. Suppose M, M i € % , 1 S i < m , and Mi+ M. Suppose E is a closed subset of S". Then

p ( M , E ) Z l i m s u p p ( M i , E ) . I + _

LEMMA 9. Suppose M, Mi E %, 1 5 i <a, and Mi + M. Suppose U i s an open subset of S". Then

p ( M , U)Sliminf p ( M i , U ) .

Let Gi and G denote the Gauss map of Mi and M,

1-50

Proof of Lemma 8: respectively. It is easy to see that G,'(E) and G-'(E) are closed sets. Set

lim G, '(E)={x€ M : x is a limit point of some sequence {x,} 1--

with x, E G;'(E)} .

To prove the assertion of Lemma 8, it suffices to show that for any open set U containing G-'(E), lirn,+- GT1(E) is a subset of U. Suppose the contrary, let x E lim,+m G;'(E)\ U. Since x & U, clearly G(x) n E = 0. However, x Elim, G;'(E) implies that x is a limit point of some sequence {x,} with X,E G;'(E). Since E is a closed subset of S", we can conclude that x E G-'(E). This contradicts the fact that G ( x ) n E = 0, so U contains lim,+- G;'(E) and the proof of Lemma 8 is then completed.

Notice that G;' (U)n GY1(S"\U) is a set of points with more than one supporting plane and hence are of zero n-dimensional measure. Thus,

Proof of Lemma 9:

lim inf p ( M , , U ) =lim inf (p(M,, S")- p ( M , , S"\U)) I - - I - -

=liminf p(M,, S")-limsup p ( M , , S"\U)

2 p ( M , S " ) - p(M, S"\U) 1-- I--

Z p ( M , U )


We introduce now briefly the concept of mixed volumes which is a natural

For K, K’ E V and t, t’ 2 0, we define and useful tool in studying the Minkowski problem.

g(t, t‘, K, K’)= V,+,(tK+ t’K‘) ,

where V,+,(tK+ t’K’) denotes the (n + 1)-dimensional measure of the set {tx + t‘x’ : x E K, X ’ E K’}. Then it is easy to see that g is a continuous function defined on [0, a) X [0, w) X % X V.

PROPOSITION 2. Let K, K’E V. Then there are non-negative constants denoted by V ( K , k ; K ’ , n + l - k ) , O s k S n + l , such that

n + l r n t l - k g(t, t ’ , K, K’) = x1 ( ) t k t V(K, k ; K’, n + 1 - k) .

k = O

Proof: We proceed by induction on the dimension. When the dimension is one, the assertion is obvious.

Suppose the proposition is true for dimension n. We are going to show that the proposition is also true for dimension n + 1. Suppose K, K’ are polyhedrons. Let K = t K + t ‘ K . Then K is also a polyhedron. Since g ( t , t ‘ , K, K’) = g(t, t ’ , K + x, K‘+ x’) for any x E K, X ’ E K‘, we may suppose K, K’ contains the origin.

Let Pl , - * * , pl be n-dimensional hyperplanes determined by the n- dimensional faces of E, and let ui be the outward normal of p,. Let PI, PI, lSiSZ, be supporting planes of K, K’ with outward normal ui. Then it is easy to verify that

(5.1) PI n K = t(Pi n K ) + t’(p: n K‘) , l S i 5 l .

Let H, H’, fi denote the support functions of K, K‘, K, respectively. By our assumption, H, H’, are all non-negative. Equation (3.4) shows that

where V , ( F i i K ) denotes the n-dimensional measure of pinK. It is easy to see that H ( u i ) = tH(ui)+ t’H’(ui). Applying (5.1) to (5.2) we obtain

1 ’ (5.3) v,+,(K) =- 1 (tH(ui) + t‘H’(ui))vn(t(Pj n K ) + tyP: n K’)) . n + l i = l


Now we see that the assertion of the proposition is valid when we apply the inductive assumption to V,,(t(P, n K) + t'(P; n K') ) . Let K , K' E % which are not necessarily polyhedrons. Let {Ki}, { K : } be polyhedrons such that K , -+ K and K: + K. Then

g(t, t', K i , KI) -+ g(t, t', K , K' ) as i -+ a

We can enclose all K i , K : in a large ball B. Therefore,

g(t, t ' , Ki, Ki) 5 g ( t , t', B, B ) = ( t + t ' ) " + l V,+,(B) .

Thus (nL1)V(Kl , k ; K ~ , n + 1 - k ) i 2 " " V n + I ( B ) , for l S i < m and 05kS

n + l . We may suppose without loss of generality that V(K,, k ; K; , n + 1 - k)

converges as i -+ 00. We then conclude that g ( t , t', K, K') can be represented as a homogeneous polynomial of degree n + 1 with non-negative coefficients. This finishes the proof of Proposition 2.

DEFINITION. V(K, k ; K ' , n + 1 - k ) is called the mixed volume of K, K' , O S k Z n + l .

We then derive an important integral formula for V(K, n; K', 1).

PROPOSITION 3. Let K, K ' E % and let H' be the support function of K'. Then

(5.4) V(K, n; K' , I)=-

Proof: Suppose K , K ' are polyhedrons and they both contain the origin. Let D ( K ' ) denote the extrinsic diameter of K'. Observe that

Let P I , * . . , P, be n-dimensional hyperplanes determined by n- dimensional faces of K and let u l , . . * , uf be their outward normals. It is obvious that

THE SOLUTION OF THE R-DIMENSIONAL MINKOWSKI PROBLEM 513

Hence,

V,,+l(K+tK')ZV,,+l(K)+ tH(u , )Vn(P inK) . i = l

On the other hand, if we define A to be the set of all points whose distance from the (n-1)-skeleton of K is less than tD(K'), then

r K + t K ' c K U A U U {x+tui : x E P i n K , 0 5 t 5 H ( u i ) } .

i = l

Consequently,

V n + l ( K + t K ' ) s V,,+,(K)+ tH(u i )V , (P inK)+O( t2 ) . i = l

Thus (5.4) is valid when K, K' are polyhedrons. The general case is then a consequence of polyhedron approximations and Proposition 1.

COROLLARY 1. Suppose Kit% and p ( K i ) converges weakly to another Bore1 measure p, on S". Then, after translations of the Ki, a subsequence of them converges to a compact conuex set K E % with p ( K ) = p.

Proof: For any U E S", let L, ={ tu : 0 5 t 5 1). For any compact convex set K'E %, V ( K , n ; L,, 1 ) is the n-dimensional measure of the projection of K' into a hyperplane with normal u. Equation (5.4) then implies that we can estimate an upper bound of V(K, , n ; L,, 1 ) for all u E S". As in Lemma 3, we can then estimate a uniform bound for the extrinsic diameter of each K,. Translate each K, so that its center of gravity locates at the origin. Since we have a uniform bound for the extrinsic diameter of each Ki, the Blaschke selection theorem guarantees that a subsequence of {Ki} will converge to some K E %. That p ( K ) = p is a direct consequence of Proposition 1.

We now state the well known Brunn-Minkowski inequality (for a simple proof, see [l], [ 5 ] ) .

THEOREM 2. Suppose K , K ' E %. Then the function h ( t )= g(t, 1 - t, K, K')''"'+'), t E [ O , 11, is a concave function in t. If K, K' do not lie on parallel hyperplanes, h ( t ) is linear if and only if K and K' are homothetic.

COROLLARY 2. Suppose K, K'E %. Then


If K, K‘ do not lie on parallel hyperplanes, equality in (5.5) holds if and only if K and K‘ are homothetic.

We can then prove the uniqueness of the Minkowski problem.

THEOREM 3. Suppose that K , K’ are elements of % and p ( K ) = p(K‘) . If p ( K , { x ~ s ” : ( x , u ) > O } ) > O for all U E S “ , then K and K’ coincide up to a translation.

Let u E S” and L, ={tu : 0 5 t 5 1). Then the support function of Proof: L,, denoted by T,, is

T(,( w ) = max (0, u - w) ,

V(K, n; L,, 1) =- IT,(w)dp(K)>O for all U E S ” .

Thus

n + l

W E S ” .

However, V(K, n; L,, 1) is the n-dimensional measure of the projection of K into a hyperplane with normal u. This shows that K does not lie on any hyperplane. Let H, H’ denote the support function of K , K’, respectively.

Then by (5.4),

V(K, n; K’, l)=- H’dp(K)=- IH’ dp(K’ ) = Vn+l(K’) n + l n + l

So an application of (5.5) shows that

Similarly, Vn+l(K)2 Vn+l(K’) and hence V,+,(K) = V,+,(K’). Therefore, (V(K, n; K‘, l))”+’ = ( Vn+l(K))”V,,+l(K’) and then K and K’ must be homothetic. However, K and K’ enclose the same volume. This shows that K and K’ coincide up to a translation.

Let K E %. Then p ( K ) will satisfy certain “compatibility” conditions.

LEMMA 10. Let K E (e. Then x, dp(K) = 0 for each coordinate function xi I on S“.

Proof: It suffices to prove the lemma for polyhedrons. The general case follows by polyhedron approximations and Proposition 1.

Suppose K is a polyhedron. Let a = (a,, - * * , an+l) be any constant vector in Rn+’. Then, since A(ZY2; aix,)=O in R”+’, we obtain, by the divergence


formula,

where n is the outward unit normal of K. Transforming (5.6) to an equation on S", we prove the assertion of the lemma.

The existence thearem of Minkowski, Alexandrov, Fenchel and Jessen shows that the converse of Lemma 10 is almost true (see [2 ] ) .

THEOREM 4. Let p be a non-negative Borel measure on S" such that

x i p = 0 for all i and p({x : (x, u ) >O}) > O for all u E S". Then there exists 1. K E % such that p = p ( K ) . Furthermore, K is unique up to translations.

Remark. As in the proof of Theorem 3, one imposes the condition p({x : (x, u>>O})>O for u E S" is to guarantee that the sought after convex body has "dimension" n + 1.

We only sketch the proof of Theorem 4. In our case, we can give a complete proof of Theorem 4 when p is

absolutely continuous. Let p be represented by a non-negative measurable function f on S". It is

easy to find positive smooth functions Ki on S" such that l/Ki + f in the L'

sense and xi(l/Ki) = 0 for 1 S j 5 n + 1, 1 5 i <a. Using Theorem 1 we can

find smooth convex bodies Mie% such that p ( M i ) is represented by l /Ki . Corollary 1 shows that there exists M E % with p ( M ) an absolutely continuous measure represented by f. This solves the Minkowski problem when p is absolutely continuous. The proof of Theorem 4 is actually analogous to the one given above. We first solve the case when p is a sum of point masses. (The solution will be a polyhedron.) For the general case, we approximate p weakly by point masses pi. Applying Corollary 1 to the solutions of pi, we obtain M E % with p ( M ) = p.

I

Bibliography

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[4] Cheng, S. Y. and Yau, S. T., On the regularity of the Monge-Ampbre equation

[S] Eggleston, H. G., Conuexity, Cambridge Univ. Press, Cambridge, 1958.

K. Jiirgens, Mich. Math. J. 5, 1958, pp. 105-126.

det (a2 u/axi ax,) = F(x, u), to appear in Comm. Pure Appl. Math., 1977.


[6] Nirenberg, L., The Weyl and Minkowski problems in differential geometry in the large, Comm.

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Pure Appl. Math., Vol. 6 1953, pp. 337-394.

#x, , . . . , x,,)>O, Soviet Math. Dokl. Vol. 12, 1971, pp. 1436-1440.

Math. Dokl. Vol. 12, 1971, pp. 1192-1196.

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on the regularity of the solution of the n-dimensional ...js/math646/cheng-yau.minkowski.pdf496...

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