THE AFFINE SOBOLEV INEQUALITY

Gaoyong Zhang 1. Introduction The Sobolev inequality is one of the fundamental inequalities connecting analysis and geometry. The literature related to it is vast (see, for example, [A1], [Be], [BH], [BL], [BP], [C], [Ch], [H], [Ho], [HV], [HS], [JL1], [K], [Li], [LZ], [R], and [Y]). In this paper, a new inequality that is stronger than the Sobolev inequality is presented. A remarkable feature of the new inequality is that it is independent of the norm chosen for the ambient Euclidean space. The Sobolev inequality in the Euclidean space Rn states that for any C 1 function f (x) with compact support there is Z n , (1.1) |∇f | dx ≥ nωn1/n kf k n−1 Rn

where |∇f | is the Euclidean norm of the gradient of f , kf kp is the usual Lp norm of f in Rn , and ωn is the volume enclosed by the unit sphere S n−1 in Rn . The best constant in the inequality is attained at the characteristic functions of balls. It is known that the sharp Sobolev inequality (1.1) is equivalent to the classical isoperimetric inequality (see, for instance, [A2], [BZ], [F], [FF], [M], [O], [SY], and [T]). We prove an affine Sobolev inequality which is stronger than (1.1). This inequality is proved by using a generalization of the Petty projection inequality to compact sets that is established in this paper (see [L1], [L2], [Gar], [Le], [S] and [Th] for the classical Petty projection inequality of convex bodies). Theorem 1.1. If f is a C 1 function with compact support in Rn , then Z 1 −n (1.2) k∇u f k1 du ≤ cn kf k−n n , n−1 n S n−1 where ∇u f is the partial derivative of f in direction is the standard surface ¡ ωnu, ¢du n measure on the unit sphere, and the constant cn = 2ωn−1 is best. The inequality (1.2) is GL(n) invariant while the inequality (1.1) is only SO(n) invariant. Thus, inequality (1.2) does not depend on the Euclidean norm of Rn . The Research supported, in part, by NSF Grant DMS-9803261 Typeset by AMS-TEX

1

2

GAOYONG ZHANG

best constant in (1.2) is attained at the characteristic functions of ellipsoids. Applying the H¨older inequality and Fubini’s theorem to the left-hand side of (1.2), one can easily see that inequality (1.2) is stronger than the Sobolev inequality (1.1). For radial functions, the inequality (1.2) reduces to (1.1). It is worth noting that the left-hand side of (1.2) is a natural geometric invariant. Specifically, for a C 1 function f (x), there is an important norm of Rn given by Z kuk = k∇u f k1 = |hu, ∇f (x)i|dx, u ∈ Rn , Rn

where h , i is the usual inner product in Rn . The volume of the unit ball of this norm is exactly the left-hand side of (1.2). We will also prove a generalization of the Gagliardo-Nirenberg inequality. n m Theorem 1.2. Let {ui }m 1 be a sequence of unit vectors in R and let {λi }1 be a sequence of positive numbers satisfying

|x|2 =

m X

λi hx, ui i2 ,

x ∈ Rn .

i=1

If f is a C 1 function with compact support in Rn , then m Y

λi

n . k∇ui f k1n ≥ 2kf k n−1

i=1

2. Basics of convex bodies A convex body is a compact convex set with nonempty interior in Rn . A convex body K is uniquely determined by its support function defined by hK (u) = maxhu, xi, x∈K

u ∈ S n−1 .

If K contains the origin in its interior, the polar body K ∗ of K is given by K ∗ = {x ∈ Rn : hx, yi ≤ 1 for all y ∈ K}. Denote by V (K) the volume of K. The mixed volume V (K, L) of convex bodies K and L is defined by V (K, L) =

1 V (K + εL) − V (K) lim+ . n ε→0 ε

There is a unique finite measure SK on S n−1 so that Z 1 V (K, L) = hL (u)dSK (u). n S n−1

THE AFFINE SOBOLEV INEQUALITY

3

The measure SK is called the surface area measure of K. When K has a C 2 boundary ∂K with positive curvature, the Radon-Nykodim derivative of SK with respect to the Lebesgue measure on S n−1 is the reciprocal of the Gauss curvature of ∂K. An important inequality of mixed volume is the Minkowski inequality, (2.1)

V (K, L)n ≥ V (K)n−1 V (L),

with equality if and only if K and L are homothetic. For a convex body K, let K|u⊥ be the projection of K onto the 1-codimensional subspace u⊥ orthogonal to u. The projection function v(K, u) of K is defined by v(K, u) = voln−1 (K|u⊥ ),

u ∈ S n−1 .

The projection function v(K, u) of K defines a new convex body ΠK whose support function is given by hΠK (u) = v(K, u) = voln−1 (K|u⊥ ),

u ∈ S n−1 .

The convex body ΠK is called the projection body of K. The volume of the polar of the projection body Π∗ K is given by 1 V (Π K) = n

Z

∗

v(K, u)−n du. S n−1

The Petty projection inequality is (see [L1] and [P]) µ (2.2)

n−1

V (K)

∗

V (Π K) ≤

ωn ωn−1

¶n ,

with equality if and only if K is an ellipsoid. It was shown in [L3] that the Petty projection inequality is stronger than the classical isoperimetric inequality of convex bodies. Let u1 , u2 , . . . , un be an orthonormal basis of Rn . For a convex body K in Rn , the Loomis-Whitney inequality is (see [LW] and [BZ], p. 95) n−1

V (K)

≤

n Y

v(K, ui ).

i=1

The Loomis-Whitney inequality was generalized by Ball [B]. Let {ui }m 1 be a sen m quence of unit vectors in R , and let {ci }1 be a sequence of positive numbers for which m X ci ui ⊗ ui = In , i=1

4

GAOYONG ZHANG

where ui ⊗ ui is the rank-1 orthogonal projection onto the span of ui and In is the identity on Rn . Then, for a convex body K in Rn , Ball proved the inequality (2.3)

n−1

V (K)

≤

m Y

v(K, ui )ci .

i=1

The condition on ui and ci is equivalent to 2

|x| =

m X

ci hx, ui i2 ,

x ∈ Rn .

i=1

For details of convex bodies, see [Gar], [S] and [Th]. 3. Inequalities for compact domains In this section, we generalize inequalities (2.1) – (2.3) to compact domains. In this paper, a compact domain is the closure of a bounded open set. The generalization of the Minkowski inequality to compact domains can be obtained from the BrunnMinkowski inequality. This appears to be standard. See [Bu] and [BZ]. If M and N are compact domains in Rn , then the Brunn-Minkowski inequality is (3.1)

1

1

1

V (M + N ) n ≥ V (M ) n + V (N ) n ,

with equality if and only if M and N are homothetic. Let M be a compact domain with piecewise C 1 boundary ∂M , and let K be a convex body in Rn . The mixed volume of M and K, V (M, K), is defined by (3.2)

1 V (M, K) = n

Z hK (ν(x))dSM (x), ∂M

where dSM is the surface area element of ∂M and ν(x) is the exterior unit normal vector of ∂M at x. If K is the unit ball Bn in Rn , then nV (M, Bn ) is the surface area S(M ) of M . Lemma 3.1. If M is a compact domain with piecewise C 1 boundary ∂M , and K is a convex body in Rn , then (3.3)

nV (M, K) = lim+ ε→0

V (M + εK) − V (M ) . ε

When M is not convex, the limit of the right-hand side of (3.3) may not exist. Equation (3.3) holds when M is a convex body or is a compact domain with piecewise C 1 boundary. We give a proof of Lemma 3.1 in the Appendix.

THE AFFINE SOBOLEV INEQUALITY

5

Lemma 3.2. If M is a compact domain with piecewise C 1 boundary, and K is a convex body in Rn , then V (M, K)n ≥ V (M )n−1 V (K),

(3.4)

with equality if and only if M and K are homothetic. Proof. For ε ≥ 0, consider the function 1

1

1

f (ε) = V (M + εK) n − V (M ) n − εV (K) n . From the Brunn-Minkowski inequality (3.1), the function f (ε) is non-negative and concave. From Lemma 3.1, we have lim+

ε→0

1−n 1 f (ε) − f (0) = V (M ) n V (M, K) − V (K) n ≥ 0. ε

This proves the inequality (3.4). If the equality holds, f (ε) must be linear. This implies that M and K are homothetic. ¤ Lemma 3.2 was proved in [Bu] when M has a C 1,1 boundary. Let M be a compact domain in Rn with piecewise C 1 boundary ∂M and exterior unit normal vector ν(x). For any continuous function f on S n−1 , define a linear functional µM on the space of continuous functions, C(S n−1 ), on S n−1 , by Z (3.5)

µM (f ) =

f (ν(x)) dSM (x), ∂M

where dSM is the surface area element of M . The linear functional µM is a nonnegative linear functional on C(S n−1 ). Since the sphere is compact, µM is a finite measure on S n−1 . The measure µM is called the surface area measure of the compact domain M . The Minkowski existence theorem states that for every finite non-negative measure µ on S n−1 so that Z (3.6)

Z |hu, vi|dµ(v) > 0, u ∈ S n−1 ,

udµ(u) = 0, and S n−1

S n−1

there exists a unique convex body K (up to translation) whose surface area measure is µ. See [S], pp. 389-393. We verify that the surface area measure µM of a compact domain M defined above satisfies (3.6). By Green’s formula, for any C 1 vector field ξ(x) in Rn , there is Z

Z hξ(x), ν(x)idSM (x) =

∂M

div ξ(x)dx. M

6

GAOYONG ZHANG

Choose ξ(x) = ei , i = 1, 2, · · · , n, the coordinate vectors, then div ξ = div ei =0. Therefore, Green’s formula yields Z hei , ν(x)idSM (x) = 0. ∂M

Let f (u) = hei , ui. Then (3.5) gives Z Z hei , uidµM (u) = S n−1

hei , ν(x)idSM (x) = 0. ∂M

Since M has non-empty interior, one has Z |hu, ν(x)i|dSM (x) > 0, ∂M

that is,

Z S n−1

|hu, vi|dµM (v) > 0.

Hence, µM satisfies the condition (3.6). Let M be a compact domain in Rn with piecewise C 1 boundary ∂M . A convexifi˘ of M is a convex body whose surface area measure S ˘ is defined by cation M M (3.7)

SM˘ = µM .

Note that a convex body is determined by its surface area measure only up to ˘ is unique up to translation. translation. Therefore, the convexification M Let M be a compact domain in Rn with piecewise C 1 boundary ∂M . The projection function v(M, u) of M on S n−1 is defined by Z 1 v(M, u) = |hu, ν(x)i|dSM (x) 2 ∂M Z 1 = |hu, vi|dµM (v), u ∈ S n−1 , 2 S n−1 where ν(x) is the exterior unit normal vector of M at x. The following lemma is obvious. ˘ is a Lemma 3.3. If M is a compact domain with piecewise C 1 boundary and M convexification of M , then ˘ , u), u ∈ S n−1 , v(M, u) = v(M ˘ , K), V (M, K) = V (M for any convex body K in Rn .

THE AFFINE SOBOLEV INEQUALITY

7

˘ Lemma 3.4. If M is a compact domain in Rn with piecewise C 1 boundary, and M is its convexification, then ˘ ) ≥ V (M ), V (M

(3.8)

with equality if and only if M is convex. Proof. From Lemma 3.3, for any convex body K, we have ˘ , K). V (M, K) = V (M ˘ . Then Let K = M ˘ ) = V (M ˘ ). V (M, M The generalized Minkowski inequality (3.4) yields ˘ ) = V (M, M ˘ ) ≥ V (M ) V (M This proves (3.8).

n−1 n

1

˘ )n . V (M

¤

The convexification in Rn was introduced in [BMP]. Lemma 3.4 for polytopes was proved in [BMP]. See also [Weil]. Let M be a compact domain in Rn with piecewise C 1 boundary ∂M . The projection body ΠM of M is defined by hΠM (u) = v(M, u),

u ∈ S n−1 .

It is easily seen that ΠM is an origin-symmetric convex body. Let `u be a line parallel to the unit vector u, and let d`u be the volume element of the subspace u⊥ orthogonal to u. Then Z 1 hΠM (u) = #(M ∩ `u )d`u . 2 This is the projection that counts (geometric) multiplicity. For the projection bodies of more general compact sets, see [SW]. The volume of the polar projection body Π∗ M is 1 V (Π M ) = n

Z

∗

v(M, u)−n du. S n−1

Note that the arithmetic average of the projection function v(M, u) over S n−1 is the surface area of M , up to a constant factor. One can view the SL(n)-invariant V (Π∗ M )−1/n as an affine surface area of M .

8

GAOYONG ZHANG

Lemma 3.5. If M is a compact domain in Rn with piecewise C 1 boundary, then µ n−1

(3.9)

V (M )

∗

V (Π M ) ≤

ωn ωn−1

¶n ,

with equality if and only if M is an ellipsoid. Proof. By (3.8), Lemma 3.3, and the Petty projection inequality (2.2) for convex bodies, we have µ n−1

V (M )

˘) ≤ ˘ )n−1 V (Π∗ M V (Π M ) ≤ V (M ∗

ωn ωn−1

¶n ,

˘ is an ellipsoid, and hence M is an with equalities if and only if M is convex and M ellipsoid. ¤ As noted earlier for convex bodies, the generalized Petty projection inequality (3.9) is stronger than the classical isoperimetric inequality for compact domains. From the H¨ older inequality, one can easily see 1+ 1

1 nωn n S(M ) ≥ V (Π∗ M )− n . ωn−1

This inequality and (3.9) imply the classical isoperimetric inequality S(M ) ≥ nωn1/n V (M )

n−1 n

.

n Lemma 3.6. Let {wi }m 1 be a sequence of non-zero vectors in R which are not contained in one hyperplane. Then for any compact domain M in Rn m Y

(3.10)

v(M, wi )λi ≥ c V (M )n−1 ,

i=1 1

where λi = hA−1 wi , wi i, c = (det A) 2 / Pm matrix given by hAx, xi = i=1 hx, wi i2 .

Qm ³ |wi | ´λi i=1

√ λi

, and A is the positive definite

Proof. First, we show the case that M is a convex body K. Let Q be a non-singular matrix so that A = QT Q, and let y = Qx. Then m m X X 2 |y| = hAx, xi = hwi , xi = λi hui , yi2 , 2

i=1 −1

where ui = λi 2 Q−T wi .

i=1

THE AFFINE SOBOLEV INEQUALITY

9

It can be easily verified that v(QT K, wi )|wi | = det(Q)v(K, Q−T wi )|Q−T wi |. From (2.3), we have n−1

V (K)

≤

m Y i=1 m Y

λi

v(K, ui )

=

m Y

v(K, Q−T wi )λi

i=1

¶λi |wi | = v(Q K, wi ) |Q−T wi | det Q i=1 µ ¶λi m Y |wi | T λi √ = v(Q K, wi ) . λ det Q i i=1 µ

T

λi

Pm Using the fact that V (QT K) = V (K) det Q and i=1 λi = n gives the inequality (3.10) when M is convex. ˘ be a convexification of M . By Lemma 3.3, M and When M is not convex, let M ˘ M have the same projection function. From (3.8) and the convex case, we have ˘ )n−1 ≤ cV (M )n−1 ≤ cV (M =

m Y i=1 m Y

˘ , wi )λi v(M v(M, wi )λi . ¤

i=1

4. The affine Sobolev inequality In this section , we prove the results stated in the Introduction. Theorem 4.1. If f is a C 1 function with compact support in Rn , then Z ¡ ωn ¢n 1 −n (4.1) k∇u f k1 du ≤ kf k−n n , n−1 n S n−1 2ωn−1 Proof. For t > 0, consider the level sets of f in Rn , Mt = {x ∈ Rn : |f (x)| > t}, St = {x ∈ Rn : |f (x)| = t}. Since f is of class C 1 , for almost all t > 0, St is a C 1 submanifold which has nonzero normal vector ∇f . Let dSt be the surface area element of St . Then one has the formula of volume elements, (4.2)

dx = |∇f |−1 dSt dt.

10

GAOYONG ZHANG

We have Z (4.3)

k∇u f k1 =

|∇u f (x)| dx Z |h∇f, ui||∇f |−1 dSt dt

n ZR∞

= 0

St

Z

∞

=2

v(Mt , u)dt. 0

On the other hand, we have (4.4)

|f |

n n−1

Rn

ÃZ

! 1 n n−1 dx = dt dx t n−1 Rn 0 µZ ¶ Z ∞ 1 n = t n−1 dx dt n−1 0 Mt Z ∞ 1 n = t n−1 V (Mt )dt. n−1 0 Z

Z

|f |

Since V (Mt ) is decreasing with respect to t, there is t

1 n−1

³ V (Mt ) = tV (Mt ) µZ

n−1 n

t

≤

V (Mτ ) 0

1 ´ n−1

n−1 n

µZ

n−1 d = n dt

V (Mt )

n−1 n

1 ¶ n−1

dτ

V (Mt )

t

V (Mτ )

n−1 n

n−1 n

n ¶ n−1

dτ

.

0

This gives Z

∞

(4.5)

t

1 n−1

0

n−1 V (Mt )dt ≤ n

µZ

∞

V (Mt )

n−1 n

n ¶ n−1 dt .

0

From (4.4) and (4.5), we obtain µZ

Z (4.6)

|f |

n n−1

Rn

dx ≤

∞

V (Mt )

n−1 n

n ¶ n−1 dt .

0

By the generalized Petty projection inequality (3.9), we obtain (4.7)

V (Mt )

n−1 n

1 ωn V (Π∗ Mt )− n ωn−1 µ Z ¶− n1 1 ωn v(Mt , u)−n du = . ωn−1 n S n−1

≤

THE AFFINE SOBOLEV INEQUALITY

11

From (4.6) and (4.7), we have n kf k n−1

ωn ≤ ωn−1

Z

∞

µ

0

Z

1 n

S n−1

¶− n1 v(Mt , u)−n du dt.

Minkowski’s inequality for integrals yields Z 0

∞

µZ S n−1

ÃZ ¶− n1 −n v(Mt , u) du dt ≤

µZ

S n−1

∞

¶−n !− n1 . v(Mt , u)dt du

0

From (4.3) and the last two inequalities, we finally obtain n kf k n−1

ωn ≤ 2ωn−1

µ

1 n

Z S n−1

−n k∇u f k1

¶− n1 du .

This proves the Theorem. ¤ We observe that inequality (4.1) is stronger than the Sobolev inequality (1.1). Indeed, the H¨older inequality and Fubini’s theorem yield µ

1 nωn

¶− n1

Z S n−1

−n k∇u f k1

≤

du

= = (4.8)

=

Z 1 k∇u f k1 du nωn S n−1 Z Z 1 |h∇f, ui|dxdu nωn S n−1 Rn Z Z 1 |h∇f, ui|dudx nωn Rn S n−1 Z 2ωn−1 |∇f |dx. nωn Rn

Inequalities (4.1) and (4.8) give the Sobolev inequality (1.1). Let us show that the generalized Petty projection inequality (3.9) can be proved by using the inequality (4.1). For compact domain M and for small ε > 0, define ½ fε (x) =

0

dist(x, M ) ≥ ε,

1−

dist(x,M ) ε

dist(x, M ) < ε.

If ε is small and dist(x, M ) < ε, then there exists a unique x0 ∈ ∂M so that dist(x, M ) = |x0 − x|. Let ν(x0 ) =

x0 − x . |x0 − x|

12

GAOYONG ZHANG

Consider

Mε = {x ∈ Rn : 0 < dist(x, M ) < ε},

¯ ε . One has and its closure M ½ ∇fε (x) = It follows that

ε−1 ν(x0 ) x ∈ Mε , ¯ ε. 0 x∈ /M Z

Z −1

Rn

|hν(x0 ), ui| dx.

|h∇fε , ui| dx = ε

Mε

Let t = dist(x, M ), 0 < t < ε. Then dx = dSM dt + o(∆t). Therefore, as ε → 0, we have Z Z −1 0 ε |hν(x ), ui| dx −→ |hν(x0 ), ui| dSM (x0 ) = 2v(M, u). Mε

∂M

On the other hand, since fε converges to the characteristic function χM of M , we have Z n |fε | n+1 dx −→ V (M ). Rn

It follows that (4.1) implies (3.9). We have seen that the affine Sobolev inequality (4.1) is equivalent to the generalized Petty projection inequality (3.9). The constant in the inequality (4.1) is sharp. It is attained at the characteristic functions of ellipsoids. 5. A generalization of the Gagliardo-Nirenberg inequality Let f be a C 1 function with compact support in Rn . Gagliardo [Gag] and Nirenberg [N] proved the inequality (5.1)

n Y

1

n . k∇xi f k1n ≥ 2kf k n−1

i=1

This inequality implies the Sobolev embedding theorem. See [A1], p. 38. We give a generalization of inequality (5.1) which is equivalent to the inequality (3.10) for compact domains. n Theorem 5.1. Let {wi }m 1 be a sequence of vectors in R not contained in one hyperplane. If f (x) is a C 1 function with compact support in Rn , then

(5.2)

m Y

λi

n , k∇wi f k1n ≥ c kf k n−1

i=1 1 ³ ´ 2n Qm where the constants λi = hA−1 wi , wi i and c = 2 det A i=1 λλi i depend only on the sequence of vectors, and A is the positive definite matrix given by hAx, xi = P m 2 i=1 hx, wi i .

THE AFFINE SOBOLEV INEQUALITY

13

Proof. Use the notations in Theorem 4.1. From (4.6), we have Z kf k From (3.10),

Pm i=1

∞

≤

V (Mt )

n−1 n

dt.

0

λi = n, and the H¨older inequality, we have

Z c

n n−1

1 n

Z

∞

V (Mt )

n−1 n

m ∞Y

dt ≤

0

0

≤

λi

v(Mt , wi ) n dt

i=1

m µZ Y

∞

¶ λni v(Mt , wi )dt ,

0

i=1

where the constant c is from (3.10). It follows that (5.3)

1 n

n c kf k n−1 ≤

m µZ Y

∞

¶ cni . v(Mt , ui )dt

0

i=1

Similar to (4.3), one has Z k∇wi f k1 = 2|wi |

∞

v(Mt , wi )dt. 0

From this and (5.3), inequality (5.2) follows. ¤ n m Corollary 5.2. Let {ui }m 1 be a sequence of unit vectors in R and let {λi }1 be a sequence of positive numbers satisfying

|x|2 =

m X

λi hx, ui i2 ,

x ∈ Rn .

i=1

If f is a C 1 function with compact support in Rn , then m Y

λi

n . k∇ui f k1n ≥ 2kf k n−1

i=1

Similar to the equivalence of (3.9) and (4.1), the geometric inequality (3.10) is equivalent to the analytic inequality (5.2). A similar argument can also be carried out. The constant of the inequality (5.2) is best. It is attained at the characteristic functions of parallelepipeds.

14

GAOYONG ZHANG

6. Appendix Proof of Lemma 3.1. Let k be a positive integer. Since ∂M is compact and of class C 1 piecewise, one can choose δ > 0 such that |hx − x0 , ν(x0 )i| ≤ k −1 |x − x0 |, |hK (ν(x)) − hK (ν(x0 ))| < k −1 ,

|ν(x) − ν(x0 )| ≤ k −1 ,

x, x0 ∈ ∂M, |x − x0 | < δ.

For x ∈ ∂M , consider a point y ∈ / M but y ∈ x + εK. We estimate the distance of 0 y to ∂M . Let x ∈ ∂M be the point which attains the distance. Then |y − x0 | = |hy − x0 , ν(x0 )i| = |hy − x, ν(x0 )i + hx − x0 , ν(x0 )i| = |hy − x, ν(x)i + hy − x, ν(x0 ) − ν(x)i + hx − x0 , ν(x0 )i| ≤ |hy − x, ν(x)i| + |y − x||ν(x0 ) − ν(x)| + |hx − x0 , ν(x0 )i|.

(6.1)

Let d be the diameter of K. Obviously, |x − x0 | < 2εd. Choose ε so that 2εd < δ. Then |hx − x0 , ν(x0 )i| ≤ k −1 2εd,

|y − x||ν(x0 ) − ν(x)| ≤ εdk −1 ,

hy − x, ν(x)i ≤ εhK (ν(x)). If hy − x, ν(x)i < 0, then |hy − x, ν(x)i| < |hx − x0 , ν(x)| ≤ k −1 2εd. When k is large enough, we have |hy − x, ν(x)i| ≤ εhK (ν(x)) ≤ εhK (ν(x0 )) +

ε . k

Therefore |y − x0 | ≤ εhK (ν(x0 )) + ε

(6.2)

3d + 1 . k

Let y 0 be a point in x + εK so that hy 0 − x, ν(x)i = εhK (ν(x)). Similar to (6.1), we have |y 0 − x0 | ≥ |hy 0 − x, ν(x)i| − |y − x||ν(x0 ) − ν(x)| − |hx − x0 , ν(x0 )i|. It follows that (6.3)

|y 0 − x0 | ≥ εhK (ν(x0 )) − ε

3d + 1 . k

THE AFFINE SOBOLEV INEQUALITY

15

Consider the regions Dε = {x : x ∈ M + εK, but x ∈ / M }, Dε± = {x + tν(x) : x ∈ ∂M, 0 ≤ t ≤ ε(hK (ν(x)) ± (3d + 1)/k}. In view of (6.2) and (6.3), we have shown Dε− ⊆ Dε ⊆ Dε+ . From the equations V (M + εK) − V (K) = V (Dε ) and V (Dε± ) lim = ε ε→0+

Z

µ ¶ 3d + 1 hK (ν(x)) ± dSM , k ∂M

we obtain the inequalities lim sup ε→0+

lim inf ε→0+

V (M + εK) − V (M ) V (Dε+ ) 3d + 1 ≤ lim = nV (M, K) + S(M ), + ε ε k ε→0 3d + 1 V (M + εK) − V (M ) V (Dε− ) ≥ lim = nV (M, K) − S(M ). ε ε k ε→0+

These prove the lemma. ¤ I would like to thank Peter McMullen and Endre Makai, Jr. who pointed out to me that convexification was introduced in [BMP], and to thank Rolf Schneider who told me that Lemma 3.2 was proved in [Bu] with a slightly different assumption.

References [A1]

T. Aubin, Nonlinear Analysis on Manifolds: Monge–Amp´ ere Equations, Springer–Verlag, Berlin, 1982. [A2] T. Aubin, Problemes isoperimetriques et espaces de Sobolev, J. Diff. Geom. 11 (1976), 1976. [BL] D. Bakry and M. Ledoux, Levy-Gromov’s isoperimetric inequality for an infinite dimensional diffusion generator, Invent. Math. 123 (1996), 259-281. K. Ball, Shadows of convex bodies, Trans. Amer. Math. Soc. 327 (1991), 891–901. [B] [Be] W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Annals Math. 138 (1993), 213–242. [BP] W. Beckner and M. Pearson, On sharp Sobolev embedding and the logarithmic Sobolev inequality, Bull. London Math. Soc. 30 (1998), 80–84. [BH] S. Bobkov and C. Houdr´ e, Some connections between isoperimetric and Sobolev-type inequalities, Memoirs of Amer. Math. Soc., Num. 616, 1997. [BZ] Yu.D. Burago and V.A. Zalgaller, Geometric Inequalities, Springer–Verlag, Berlin, 1988. [BMP] K. Boroczky, I. B´ ar´ any, E. Makai Jr., and J. Pach, Maximal volume enclosed by plates and proof of the chessboard conjecture, Discrete Math. 69 (1986), 101–120. [Bu] H. Busemann, The isoperimetric problem for Minkowski area, Amer. J. Math. 71 (1949), 743–762.

16 [C]

GAOYONG ZHANG

E. Carlen, Superadditivity of Fisher’s information and logarithmic Sobolev inequalities, J. Funct. Anal. 101 (1991), 194–211. [Ch] S.Y.A. Chang, The Moser-Trudinger inequality and applications to some problems in conformal geometry, Lecture Notes, Summer Geometry Institute at Park City, Utah, 1992. [F] H. Federer, Geometric Measure Theory, Springer-Verlag, Berlin, 1969. [FF] H. Federer and W. Fleming, Normal and integral currents, Ann. Math. 72 (1960), 458–520. [Gag] E. Gagliardo, Properiet´ a di alcume classi di funzioni in piu variabili, Ric. Mat. 7 (1958), 102–137. [Gar] R.J. Gardner, Geometric Tomography, Cambridge University Press, Cambridge, 1995. [Grom] M. Gromov, Isoperimetric inequalities in Riemannian manifolds, In: Asympototic Theory of Finite Dimensional Normed Spaces (by V.D. Milman and G. Schechtman), Lecture Notes in Math. vol. 1200, Springer-Verlag, Berlin, 1986. [Gros] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), 1061–1083. [H] R. Hamilton, An isoperimetric estimate for the Ricci flow on the two-sphere, Modern methods in complex analysis (Princeton, NJ, 1992), 191–200, Ann. of Math. Stud., 137, Princeton Univ. Press, Princeton, NJ, 1995. [HV] E. Hebey and M. Vaugon, The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds, Duke Math. J. 79 (1995), 235–279. [HS] D. Hoffman and J. Spruck, A Sobolev inequality for Riemannian submanifolds. Differential geometry, (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 1, pp. 139–141. Amer. Math. Soc., Providence, R.I., 1975. [Ho] R. Howard, The sharp Sobolev inequality and the Banchoff-Pohl inequality on surfaces, Proc. Amer. Math. Soc. 126 (1998), 2779-2787. [JL1] D. Jerison and J. Lee, Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc. 1 (1988), 1–13. [JL2] D. Jerison and J. Lee, The Yamabe problem on CR manifolds, J. Diff. Geom. 25 (1987), 167–197. [K] S. Klainerman, Remarks on the global Sobolev inequalities in the Minkowski space Rn+1 , Comm. Pure Appl. Math. 40 (1987), 111–117. [Le] K. Leichtweiss, Affine geometry of convex bodies, Johann Ambrosius Barth, Heidelberg, 1998. [Li] P. Li, On the Sobolev constant and the p-spectrum of a compact Riemannian manifold, Ann. Sci. cole Norm. Sup. 13 (1980), 451–468. [LZ] Y.Y. Li and M. Zhu, Sharp Sobolev inequalities involving boundary terms, Geom. Funct. Anal. 8 (1998), 59–87. [LW] L.H. Loomis and Whitney, An inequality related to isoperimetric inequality, Bull. Amer. Math. Soc. 55 (1949), 961–962. [L1] E. Lutwak, On some affine isoperimetric inequalities, J. Diff. Geom. 23 (1986), 1–13. E. Lutwak, Selected affine isoperimetric inequalities, In: Handbook of Convex Geometry [L2] (P.M. Gruber and J.M. Wills Eds) Vol. A, Elsevier Science Publishers, North-Holland, 1993, 151–176. [L3] E. Lutwak, A general isepiphanic inequality, Proc. Amer. Math. Soc. 90 (1984), 415–421. V.G. Maz’ya, Classes of domains and imbedding theorems for function spaces, Dokl. Akad. [M] Nauk. SSSR 133 (1960), 527–530. [N] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. 13 (1959), 116–162. [O] R. Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), 1182–1238. [P] C.M. Petty, Isoperimetric problems, Proc. Conf. Convexity and Combinatorial Geometry, Univ. of Oklahoma, Norman, 1971, 26–41. [R] O.S. Rothaus, Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities, J. Funct. Anal. 64 (1985), 296–313. [S] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge, 1993.

THE AFFINE SOBOLEV INEQUALITY [SW]

[SY] [T] [Th] [Weil] [Weis] [Y]

17

R. Schneider and J. Wieacker, Integral geometry, In: Handbook of Convex Geometry (P.M. Gruber and J.M. Wills Eds) Vol. A, Elsevier Science Publishers, North-Holland, 1993, 1349– 1390. R. Schoen and S.T. Yau, Lecturers on Differential Geometry, International Press, Boston, 1994. G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura. Appl. 110 (1976), 353–372. A.C. Thompson, Minkowski Geometry, Cambridge University Press, Cambridge, 1996. W. Weil, mean bodies associated with random closed sets, Rendiconti del Circolo Matematico Di Palermo, Serie II, Suppl. 50 (1997), 387-412. F. Weissler, Logarithmic Sobolev inequalities for the heat-diffusion semi-group, Trans. Amer. Math. Soc. 237 (1978), 255–269. S.T. Yau, Sobolev inequality for measure space, Tsing Hua lectures on geometry and analysis (Hsinchu, 1990–1991), 299–313, Internat. Press, Cambridge, MA, 1997.

Department of Mathematics Polytechnic University 6 MetroTech Center Brooklyn, NY 11201 [email protected]