Rate of convergence of geometric symmetrizations

Transcription

1 Rate of convergence of geometric symmetrizations B. Klartag School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel Abstract It is a classical fact, that given an arbitrary convex body K R n, there exists an appropriate sequence of Minowsi symmetrizations or Steiner symmetrizations), that converges in Hausdorff metric to a Euclidean ball. Here we provide quantitative estimates regarding this convergence, for both Minowsi and Steiner symmetrizations. Our estimates are polynomial in the dimension and in the logarithm of the desired distance to a Euclidean ball, improving previously nown exponential estimates. Inspired by a method of Diaconis [D], our technique involves spherical harmonics. We also mae use of an earlier result by the author regarding isomorphic Minowsi symmetrization. Introduction Let K R n be a convex body, and denote by and, the usual Euclidean norm and scalar product in R n. Given a vector u S n = {x R n ; x = }, we denote by π u x) = x x, u u the reflection operator with respect to the hyperplane through the origin, which is orthogonal to u in R n. The result of a Minowsi symmetrization sometimes called Blasche symmetrization) of K with respect to u, is the body τ u K) = K + π uk) where the Minowsi sum of two sets A, B R n is defined as A + B = {a + b; a A, b B}. Let h K denote the supporting functional of K, i.e. for u R n h K u) = sup x, u. x K Supported in part by the Israel Science Foundation and by the Minowsi center for Geometry.

2 Then h τuk)v) = [h Kv) + h K π u v))]. The mean width of K is defined as wk) = M K) = h S n K u)dσu), where σ is the unique rotation invariant probability measure on the sphere. The mean width is preserved under Minowsi symmetrizations. Steiner symmetrization of K with respect to a hyperplane H yields the unique body S H K) such that for any line l perpendicular to H, i) S H K) l is a closed segment whose center lies on H. ii) MeasK l) = MeasS H K) l). where Meas is the one dimensional Lebesgue measure in the line l. Steiner symmetrization preserves the volume of a set and transforms convex sets to convex sets. See e.g. [BF] for more information about these symmetrizations, and their applications in proving geometric inequalities. Consecutive Minowsi/Steiner symmetrizations may cause a convex body to resemble a Euclidean ball. Starting with an arbitrary convex body, one may apply a suitable sequence of Minowsi/Steiner symmetrizations, and obtain a sequence of bodies that converges to a Euclidean ball. This Euclidean ball would have the same mean width/volume as had the original body. In this note, we investigate the rate of this convergence. We as how many symmetrizations are needed, in order to transform an arbitrary convex body K R n into a body that is ε-close to a Euclidean ball. Our question is almost isometric in its nature, as we try to provide reasonable estimates even for small values of ε. Previous results in the literature are mostly of isomorphic nature, in the sense that the symmetrization process is aimed at obtaining a body which is uniformly isomorphic to a Euclidean ball a body is isomorphic to a Euclidean ball if its distance to a Euclidean ball is bounded by some fixed, universal constant). The first quantitative result regarding Minowsi symmetrization appears in [BLM]. Denote by D the standard Euclidean ball in R n. Their result reads as follows: Theorem. Let 0 < ε <, n > n 0 ε). Given an arbitrary convex body K R n, there exist cn log n + cε)n Minowsi symmetrizations that transform K into a body K such that ε)m K)D K + ε)m K)D where cε), n 0 ε) are of the order of expcε log ε ) and c > 0 is a numerical constant. Their proof uses the method of random Minowsi symmetrizations. In [K], the notion of randomness was altered, and has lead to an improvement of the dependence on the dimension n. The following is proved in [K]:

3 Theorem. Let n and let K R n be a convex body. Then there exist 5n Minowsi symmetrizations, such that when applied to K, the resulting body K satisfies, ) log log n c M K)D K log n where c > 0 is some numerical constant. ) log log n + c M K)D log n Note that both in [K] and in [BLM], for any fixed dimension, one cannot even formally conclude that there is convergence to a Euclidean ball. This note fills that gap in the literature, and also provides surprisingly good dependence on ε. The following theorem is proved here: Theorem.3 Let n, 0 < ε <, and let K Rn be a convex body. Then there exist cn log ε Minowsi symmetrizations, that transform K into a body K that satisfies ε)m K)D K + ε)m K)D where c > 0 is some numerical constant. Our approach to the problem of Minowsi symmetrization involves a number of novel ideas. First, rather than applying random Minowsi symmetrizations, at each step we apply n symmetrizations with respect to the vectors of some random orthonormal basis. This change of randomness improves the rate of convergence by a factor of log n see [K], [K] and also the remar following Corollary 3.3 here). Second, the use of spherical harmonics allows us to obtain good estimates regarding symmetrization of polynomials on the sphere. Finally, we approximate the supporting functional of K with an appropriate polynomial applying Theorem. and a Jacson type theorem), and use the estimates obtained for symmetrization of polynomials. Quantitative estimates regarding Steiner symmetrization are more difficult to obtain, as the problem is non-linear. The earliest estimate in the literature is due to Hadwiger [H]. It gives an estimate of the order of c n ε ) n for the number of Steiner symmetrizations required in order to transform an arbitrary n-dimensional convex body, to become ε-close to a Euclidean ball. In addition, an isomorphic result appears in [BLM], which was improved by a logarithmic factor in [KM]. The following is proved in [KM]: Theorem.4 Let n and let K R n be a convex body, with V olk) = V old). Then there exist 3n Steiner symmetrizations, such that when applied to K, the resulting body K satisfies, cd K CD 3

4 where c, C > 0 are some numerical constants. Some related estimates also appear in [T]. Our result is the first estimate which is polynomial in n and in log ε. This shows that the precise geometric shape of a convex body cannot prevent fast symmetrization of the body into an almost Euclidean ball. In this note we shall prove the following theorem. Theorem.5 Let K R n be a convex body, and let 0 < ε <. Let r > 0 be such that V olk) = V olrd). Then there exist cn 4 log ε Steiner symmetrizations, that transform K into a body K that satisfies ε)rd K + ε)rd where c > 0 is some numerical constant. The powers of n and log ε in Theorem.5 seem non optimal. We conjecture that cn log ε Steiner symmetrizations are sufficient. Regarding Minowsi symmetrizations, our result is tight in the sense that the powers in Theorem.3 cannot be improved. The proof of Theorem.5 is an application of Theorem.3 and of a geometric result by Boowsi and Heil. Throughout this paper, we denote by c, C, c etc. positive numerical constants whose value is not necessarily equal in different appearances. Spherical Harmonics In this section we summarize a few facts about spherical harmonics, to be used later on. For a comprehensive discussion on the subject, we refer the reader to the concise expositions in [SW], chapter IV., in [M] and in [G]. P : R n R is a homegeneous harmonic of degree, if P is a homogeneous polynomial of degree in R n, and P is harmonic i.e. P 0). We denote, S = {P S n ; P : R n R is a homogenous harmonic of degree } where P S n is the restriction of the polynomial P to the sphere. S is the space of spherical harmonics of degree. It is a linear space of +n ) n+ 3)! dimension! n )!. For, the spaces S and S are orthogonal to each other in L S n ). In addition, if P is a polynomial of degree in R n, then P S n can be expressed as a sum of spherical harmonics of degrees not larger than. Therefore, L S n ) = S. Spherical harmonics possess many symmetry properties, partly due to their connection with the representations of On) e.g. [V], chapter 9). For a fixed dimension n, the Gegenbauer polynomials {G i t)} i=0 are defined by the following three conditions: 4

5 i) G i t) is a polynomial of degree i in one variable. ii) For any i j we have G it)g j t) t ) n 3 dt = 0. iii) G i ) = for any i. The Gegenbauer polynomials are closely related to spherical harmonics. Next, we reformulate Lemma from [G], which is credited to Schneider. This useful lemma also follows from Corollary.3, chapter IV of [SW], and is true for all n. Lemma. Let g S be such that g = g x)dσx) =. S n Then, gu x)gu y)dµu) = G x, y ) On) where µ is the Haar probability measure on On). The following lemma reflects the fact that S is an irreducible representation space of On). We denote by P roj S : L S n ) S the orthogonal projection onto S. Lemma. Let f L S n ), and let g S be such that g =. Then, ) fux)gx)dσx) dµu) = P roj S f) ) On) S dims n ) where µ is the Haar probability measure on On). Proof: Let {g,.., g N } be an orthonormal basis of S. Then, dims ) = i= On) On) ) fux)g i x)dσx) dµu) ) S n P roj S f U) dµu) = P roj S f) because of the rotation invariance of S. Therefore, it is sufficient to prove that the integral in ) does not depend on the choice of g S, as long as it satisfies g =. Indeed, in that case each of the summands in ) equals P roj S f) dims ), for an arbitrary orthonormal basis {g,.., g N } of S. Let us try to simplify the integral in ): fux)gx)dσx) fuy)gy)dσy)dµu) On) S n S n = fx)fy) gu x)gu y)dµu)dσx)dσy). S n S n On) 5

6 By Lemma., On) gu x)gu y)dµu) = G x, y ). Hence, the integral in ) equals fx)fy)g x, y )dσx)dσy) S n S n which does not depend on g, and the lemma is proved. 3 Spherical Harmonics and Minowsi Symmetrization In this section we apply a series of Minowsi symmetrizations to a convex body K R n. Each step in the symmetrization process consists of symmetrizing K with respect to the n vectors of an orthonormal basis {e,.., e n } in R n. Such a step is denoted here as an orthogonal symmetrization with respect to {e,.., e n }. Applying an orthogonal symmetrization with respect to {e,.., e n } to K, yields a body denoted by K. Let h be the supporting functional of K, and h be the supporting functional of K. Then, n ) h x) = E ε h ε i x, e i e i 3) where the expectation is over ε {±} n, with respect to the uniform probability measure on the discrete cube. Note that by 3), orthogonal symmetrization may be viewed as an operation on support functions, rather than on convex bodies. Furthermore, we may apply an orthogonal symmetrization to any function on the sphere, which is not necessarily a support function of a convex body. Next, we analyze the effect of orthogonal symmetrizations on spherical harmonics. Let be a positive integer. A function g L S n ) is called invariant with respect to the orthonormal basis {e,.., e n }, if for any ε {±} n, we have gx) = g i ε i x, e i e i ). For a fixed orthonormal basis {e,.., e n } in R n, we denote by S 0 the linear space of all invariant functions in S. Let P roj S 0 : S S 0 be the orthogonal projection in L S n ). Then for g S, n ) g x) = E ε g ε i x, e i e i g = P roj S 0 g), i= i.e. the orthogonal symmetrization of g is the projection of g onto S 0. i= Lemma 3. If is odd, dims 0 ) = 0. Otherwise, dims) 0 n + = ). n 6

7 Proof: The odd case is easy, since for g S we necessarily have gx) = g x), and for g SK 0 we have gx) = g x). Hence, only 0 S 0. Next, assume that is even, and let g S0 be an invariant polynomial with respect to the basis {e,.., e n }. We use the coordinates x,.., x n with respect to this basis. Fixing x,.., x n the polynomial g satisfies g x,..,x n x ) = g x,..,x n x ), and hence only even degrees of x occur in g x,..,x n. By repeating the argument for the rest of the variables, we get that g is a function of x,.., x n alone. We can write, / gx,.., x n ) = x j n A j x,.., x n ) 4) j=0 where A j is a homogeneous polynomial of degree j, which depends solely on x,.., x n. Let us calculate the Laplacian of 4): / 0 = j= / jj )x j n A j x,.., x n ) + j=0 or equivalently, g S 0 if and anly if for all 0 j, x j n A j x,.., x n ) j + )j + )A j+ = A j. 5) Therefore we are free to choose A 0 any way we lie, as long as it is a homogeneous polynomial of degree, which involves only even powers of the n variables. When A 0 is fixed, A, A etc. are determined by equation 5), and the function g is recovered. Hence, dims 0 ) equals the dimension of the space of the possible A 0 x,.., x n ), which is the dimension of the space of all homogeneous polynomials of degree / in n variables. This number is nown to n + be ). n ) n + n+ We denote N = dims ) = n n+, and for an even ) n + / denote N 0 = dims0 ) =. Clearly, these two quantities n depend on n which is absent from the notation, yet the appropriate value of n will be obvious from the context. We are now ready to calculate the L norm of a random orthogonal symmetrization of a spherical harmonic - an orthogonal symmetrization with respect to a basis that is chosen uniformly over On). Clearly, any orthogonal symmetrization of an odd degree spherical harmonic vanishes. The even case is treated in the following proposition. 7

8 Proposition 3. Let be a positive even integer, and let g S be a spherical harmonic. We randomly select an orthonormal basis {v,.., v n } On), and symmetrize g with respect to this basis. Then, E g v,..,v n = N 0 g < N n + ) / g where the expectation is over the random choice of {v,.., v n } On) with respect to the Haar probability measure on On)). Proof: Fix an orthonormal basis {e,.., e n } of R n, and consider S 0 with respect to that basis. Fix also an orthonormal basis S,.., S N 0 of. From the discussion before Lemma 3., S 0 g e,..,e n = P roj S 0 g) and if the columns of U On) are {v,.., v n }, then ) g v,..,v n = P roj S 0 g U) U. Hence, N 0 g v,..,v n = P roj S 0 g U) = gux)s j x)dσx) S n and by Lemma., Note that N 0 = n + / N n + i= E g v,..,v n = j= N 0 j= g = N 0 g N N. / n + i )/ + i) n + i 3)n + i ) < / i= ) / + i / + i + n ) / ) /. which lies between n +/ and n + ) / ) / Since n + is a decreasing function of, then n + for any, and we obtain the following corollary: n Corollary 3.3 Let f L S n ) satisfy S n fx)dσx) = 0. We randomly select {v,.., v n } On). Then, E f v,..,v n < c n f where the expectation is taen over the choice of {v,.., v n } On), and c =. 8

9 Proof: Expand f into spherical harmonics: f = = f where f = P roj S f). Then f v,..,v n = = f ) v,..,v n and E f v,..,v n = = E f ) v,..,v n n f n f. An application of Jensen inequality concludes the proof. Remar: Using similar methods, one can prove that if g S and τ u g)x) = gx)+gπux)), then E u τ u g) = n + n + g. Note the advantage of symmetrizing with respect to the n vectors of a random orthonormal basis, compared to symmetrization with respect to n random sphere vectors. For instance, if = then ) n n + n + e. Hence n random symmetrizations may reduce the expectation of the L norm only by a constant factor. 4 Decay of L norm In Proposition 3. and Corollary 3.3 we established a sharp estimate for the decay of the L norm under an orthogonal symmetrization. Now we deal with the more difficult problem of estimating the decay of the L norm of the function. Our main tool is the following nown lemma see e.g. page 4 of [M]): Lemma 4. Let g S be a spherical harmonic of degree. Then, where g = sup x S n gx). g dims ) g = N g We mae use of the following well-nown estimate of binomial coefficients. For any n, n ) n < e ) n ). 6) In the following combinatorial lemmas, log is to be understood as the natural logarithm. 9

10 Lemma 4. Let ε > 0, n 3, and let be an integer. Then, +log c + ε) +log+ N n) > n ε 3 where c > 0 is some numerical constant. Proof: Denote α = /n. Case : α <. In this case, + log + n) < 3, and for c > 9, +log c + ε) +log+ N n) > N 3+3 log+ ε) > N N 3 log+ ε) > N ε 3 n ε 3 since for we always have N n 3. Case : α. In this case, + log + n ) n. 6), N > For c > 6, n+ n ) ) < log + n. By +log+ c ε) +log+ N n) > + ) ) 3+3 log + ε) n log+ n ) n for any n 3. = e 3n ) + ) 3n ) > n ε ε 3 Lemma 4.3 Let n 3, and let = αn > 0 be an even number. Then, ) N 0 T < N N for T = c [ + log + α)], where c > 0 is a numerical constant. Proof: Since n+ n+ >, it is sufficient to prove that ) n + / / ) n + Case : α <. The left hand side of 7) is equal to: / / + i / + i + n i= T T < 0 < ). 7) n + + n ) T ) T. n

11 To obtain 7) it is enough to prove that ) T n < e + n according to 6). Now, because α = n <, for T = 8, ) T n < ) ) T ) < n 3e ) m Case : α. Since = l equals: n i= ) T / + i < + i ) ) < n e + n ). ) m, the left hand side of 7) also m l ) n )T n + / < n + ) n )T 5 6 since n < and because x+/ x+ is an increasing function of x. Now, for any T > +log+ n ) log6/5), ) n )T ) n ) 5 n < < 6 e n + n ) log6/5) Since for n 3, we have +log+ is proved. n + n ). < 0 [ + log + α)], the lemma 5 Proof of the Minowsi symmetrization result We mae use of Jacson s theorem for the sphere, due to Newman and Shapiro [NS]: Theorem 5. Let n, > 0 be integers, and let f : S n R be a λ-lipschitz function on the sphere i.e. fx) fy) λ x y for any x, y S n ). Then there exists a polynomial P of degree in n variables, such that for any x S n, fx) P x) c 3 λ n where c 3 > 0 is some numerical constant.

12 Proof of Theorem.3: We assume that M K) =. Begin with 5n symmetrizations, according to Theorem., to obtain a centrallysymmetric body K. Denote by h its supporting functional. Then h is a norm and hence its Lipschitz constant equals sup x S n hx). By Theorem., log log n sup hx) < + c < c 4 8) x S n log n for some numerical constant c 4 > 0. Hence h is a c 4 -Lipschitz function, and by Theorem 5., there exists a polynomial P ε x) of degree = n ε such that, sup P ε x) hx) < c 4 c 3 ε. 9) x S n Let P ε x) = i=0 P ix) be the expansion of P ε into spherical harmonics. Randomly select T orthonormal bases i.e. the bases are chosen independently and uniformly in On)). Apply the corresponding T orthogonal symmetrizations to P ε and P,.., P, to obtain the random polynomials P ε and P,.., P. Note that still P ε = i=0 P i. Successive application of Proposition 3. yields that for an even i > 0, ) N E P i 0 T = i P i. Combining this with Lemma 4. assume n 3), ) N E P i 0 T N i i P i N. i N i [ )] Assume that T > c + )c + log + ε. According to Lemma 4.3, +log+ ) c+) ε) E P i +log+ < N n) i i Pi N i +log+ c ε) +log+ < N n) i i P i < ε3 n P i where the last inequality follows from Lemma 4.. Denote I = P 0 = P S n ε x)dσx). Then, E P εx) I E P ix) E P i x) n ε i= i= i= ε 3 n P ix) < ε P ε < ε P ε < εc 4 + c 4 c 3 ε)

13 where the last inequality follows from 8) and 9). Apply the same T orthogonal symmetrizations to K, and obtain K. Denote by h the supporting functional of K. Then, sup h x) P εx) < c 4 c 3 ε, x S n and since by 9) we have c 4 c 3 ε < I < + c 4 c 3 ε, then Clearly, E sup h x) < c 4 c 3 ε + c 4 c 3 ε + εc 4 + c 4 c 3 ε) < c ε. x S n sup h x) < c ε c ε)d K + c ε)d. x S n [ )] To summarize, we applied 5n+c +)c + log + ε n Minowsi symmetrizations to an arbitrary convex body, some of which were chosen randomly. As a result of these symmetrizations, we obtained a body such that the expectation of its distance to a Euclidean ball is no more than c ε. Therefore, there exists some numerical constant c > 0, and cn log ε symmetrizations that bring the body to be ε-close to a Euclidean ball. Remars:. The case n = should be treated separately. In this case, dims ) =, dims ) = for any. It is easy to verify that the proof wors in this case as well.. Theorem.3 is optimal in the sense that one cannot obtain an estimate for the number of minimal symmetrizations, of the form fn)gε) with fn) << n or gε) << log ε. Indeed, the dependence on n should be at least linear, as it taes a segment n symmetrizations just to become n-dimensional. Regarding the dependence on ε, if we tae a segment and apply any c log ε symmetrizations, then the segment is transformed into a zonotope which is a sum of no more than ε segments. Even c in dimension two, this zonotope cannot be ε-close to a Euclidean ball, for a small enough c. 3. Note that Theorem.3 is not tight for all possible values of n and ε. For example, Theorem. is better than Theorem.3 when log log n ε = c log n. 6 Application to Steiner Symmetrization In this section we prove Theorem.5. We mae use of a result due to Boowsi and Heil. The following theorem is a special case of Theorem in [BH] the case i, j, ) = 0, d, d) in the notations of that paper). 3

14 Theorem 6. Let K RD be a convex body. Then, n R n M K) V olk) V old) + n )R n. An immediate corollary follows: Corollary 6. Let ε > 0, and let K + ε)d be a convex body in R n with V olk) = V old). Then, M K) < + n ) ε. In addition, if ε < n then, M K) < + ) ε. n Proof : By Theorem 6., since V olk) V old) =, M K) + ε) n ) + n < + ε) + ε) n n ) + n and therefore M K) < + n ) ε. Now, assume that ε < Using the elementary inequality +ε) n we obtain M K) + ε) n ) + n [ n )ε + n. < n )ε + nn ) ε, ] nn ) ε < + ε ε n + ε < + ε ε n. Given a convex body K R n, define RK) = inf{r > 0; K RD}. Lemma 6.3 Let K R n be a convex body with V olk) = V old). Assume that there exists 0 < ε < C such that RK) = + ε, where C >. Then there exist c 5 nlog ε + log n) Steiner symmetrizations that transform K into K such that R K) < + ) n ε and if ε < n, R K) < + ) ε 4n where c 5 = c 5 C) > 0 depends solely on C. 4

15 Proof: Let K be the body obtained from K after the cn log 4Cn3 ε symmetrizations given by Theorem.3. Despite the fact that Theorem.3 is concerned with Minowsi symmetrizations, we apply the corresponding Steiner symmetrization with respect to the same hyperplanes). Since Steiner symmetrizations are contained in Minowsi symmetrizations, R K) < + ε ) 4Cn 3 M K). Apply corollary 6. and the fact that ε < C to get that R K) < + ε ) [ 4Cn 3 + n ) ] ε < + and if ε < n, R K) < n ) ε + ε ) [ 4Cn 3 + ) ] ε < + ) ε. n 4n Proposition 6.4 Let n, 0 < ε <, and let K Rn be a convex [ body with V olk) = V old). Then there exist c 6 n 3 log n + n log ] ε Steiner symmetrizations, that transform K into K which satisfies R K) < + ε where c 6 > 0 is a numerical constant. Proof: First, apply 3n Steiner symmetrizations to K, according to Theorem.4, to obtain an isomorphic Euclidean ball K. Then, K CD. Let us define a sequence of convex bodies: K 0 = K, and K i is obtained from K i using c 5 nlog RK i ) + log n) Steiner symmetrizations, as in Lemma 6.3. Then, RK i ) < ) n [RK i ) ] < ) i n [RK 0 ) ]. 0) Let T be the minimal integer such that RK T ) < + n. Since RK 0 ) < C, then by 0) necessarily T < cn log n. For any i T we have RK i ) + n and hence by Lemma 6.3 we used 5

16 no more than c n log n symmetrizations to obtain K i from K i. In total, we used less than cn 3 log n symmetrizations to obtain K T. By Lemma 6.3 for any i > 0, RK T+i) < ) [RK T+i ) ] < ) i. 4n 4n Let T be the first integer such that RK T+T ) < + ε. Then T < cn log ε. Define K = K T+T. For any T < i T + T we used no more than c nlog ε + log n) symmetrizations to obtain K i from K i. In total we applied a maximum of cn 3 log n + cn log ε Steiner symmetrizations. Proposition 6.4 proves the existence of a rather small circumscribing ball for the symmetrized body. In order to symmetrize the body from below, we use the following standard lemma. Its proof is outlined for completeness. Lemma 6.5 Let 0 < ε <, and let K R n be a convex body with M K). Assume that K [ + c 7 ε) n ] D. Then ε)d K where c 7 > 0 is some numerical constant. Proof: Assume on the contrary that there exists x 0 S n with x 0 < ε, where = h K ). Then for x S n with x x 0 <, we have ε 4 x x 0 + x x 0 < ε + + c 7 ε) n ) x x 0 < ε. Denote A = {x S n ; x x 0 ε 4 }. Then, M K) = x dσx) < σa)) + c 7 ε) n ) + σa) ε ). S n The projection of A onto the hyperplane orthogonal to x 0 contains a Euclidean ball of radius larger than ε 4. Therefore, σa) > V old ) n n ) ε V ols n ) 4 > ) n ε ε ) n πn 4 > 30 where D n is the n dimensional Euclidean unit ball, and V ol is interpreted here as the n dimensional volume. Thus, M K) < + c 7 ε) n σa) ε ε ) n < + c 7ε) n 30 6

17 and for c 7 = 30 we obtain a contradiction. Proof of Theorem.5: It is sufficient to consider the case V olk) = V old). Apply Proposition 6.4 with ε = c 7 ε) n. We use [ ] c 6 n 3 log n + n log ε < c n 4 log ε Steiner symmetrizations, and obtain a body K such that K + c 7 ε) n ) D. Since V olk) = V old), by Urysohn M K). Using Lemma 6.5, and the theorem is proved. ε)d K + c 7 ε) n ) D + ε)d Remar: Theorem. is crucial to the proof of Theorem.3. Only after obtaining the precise isomorphic statement regarding Minowsi symmetrization, can we prove the sharp almost isometric version. However, in the proof of Theorem.5 we may apply weaer estimates than that in Theorem.4, and derive the same conclusion. This could be another indication that the powers in Theorem.5 are not optimal. Acnowledgement. Part of the research was done during my visit to the University of Paris V I, and I am grateful for their hospitality. Thans also to the anonymous referee for the thorough review of the paper. References [BH] Boowsi J., Heil E., Integral representations of quermassintegrals and Bonnesen-style inequalities. Arch. Math., Vol. 47, No. 986) [BF] Bonnesen T., Fenchel W., Theorie der onvexen Körper. Springer, Berlin 934); English transl. Theory of convex bodies, BCS associates 987). [BLM] Bourgain J., Lindenstrauss J., Milman V.D., Minowsi Sums and Symmetrizations. Geometric Aspects of Functional Analysis - Israel Seminar ), Lindenstrauss J., Milman V.D. Eds.), Springer LNM, Vol ) [BLM] Bourgain J., Lindenstrauss J., Milman V.D., Estimates related to Steiner symmetrizations. Geometric aspects of functional analysis ), Lecture Notes in Math., vol. 376, Springer Berlin, 989)

18 [D] [G] [H] Diaconis P., Finite Fourier methods: access to tools. Probabilistic combinatorics and its applications San Francisco, CA, 99), Proc. Sympos. Appl. Math., 44, Amer. Math. Soc., Providence 99) Groemer H., Geometric applications of Fourier series and spherical harmonics. Encyclopedia of Math. and its Applications, 6, Cambridge Univ. Press, Cambridge 996). Hadwiger H., Einfache Herleitung der isoperimetrischen Ungleichung für abgeschlossene Puntmengen. Math. Ann. 4 95), [K] Klartag B., Remars on Minowsi Symmetrizations. Geometric Aspects of Functional Analysis - Israel Seminar ), Milman V.D., Schechtman G. Eds.), Springer LNM, Vol ) [K] Klartag B., 5n Minowsi symmetrizations suffice to arrive at an approximate Euclidean ball. Annals of Math., Vol. 56, No. 3 00) [KM] Klartag B., Milman V.D., Isomorphic Steiner Symmetrizations. Invent. Math., Vol. 53, No ) [M] Müller C., Spherical Harmonics. Springer Lecture Notes in Math., Vol ). [NS] Newman D.J., Shapiro H.S., Jacson s theorem in higher dimensions. On Approximation Theory Proceedings of a conference in Oberwolfach, 963), Birhäuser, Basel 964) [SW] Stein E.M., Weiss G., Introduction to Fourier analysis on Euclidean spaces. Princeton Mathematical Series, No. 3. Princeton University Press, Princeton, N.J. 97). [T] [V] Tsolomitis A., quantitative Steiner/Schwarz-type symmetrization. Geom. Dedicata, Vol. 60, no. 996) Vilenin N.J., Special Functions and the Theory of Group Representations. Translations of Math. Monographs, Amer. Math. Soc., Vol., Providence, Rhode Island 968). 8