Inequalities for the surface area of projections of convex bodies

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1 Iequalities for the surface area of projectios of covex bodies Apostolos Giaopoulos, Alexader Koldobsky ad Petros Valettas Abstract We provide geeral iequalities that compare the surface area S(K) of a covex body K i R to the miimal, average or maximal surface area of its hyperplae or lower dimesioal projectios. We discuss the same questios for all the quermassitegrals of K. We examie separately the depedece of the costats o the dimesio i the case where K is i some of the classical positios or K is a projectio body. Our results are i the spirit of the hyperplae problem, with sectios replaced by projectios ad volume by surface area. Itroductio The startig poit of this article are two iequalities of the secod amed author about the surface area of hyperplae projectios of projectio bodies. I [0] it was proved that if Z is a projectio body i R the (.) Z mi ξ S S(P ξ (Z)) b S(Z), where S(A) deotes the surface area of A, P ξ (Z) stads for the orthogoal projectio of Z to the hyperplae ξ perpedicular to a vector ξ S, ad (.2) b = ( )ω, ω where ω m is the volume of the Euclidea uit ball B2 m i R m. Note that b ad that (.) is sharp; there is equality if Z = B2. Coversely, i [] it was proved that if Z is a projectio body i R which is a dilate of a body i isotropic positio, the (.3) Z max ξ S S(P ξ (Z)) c(log ) 2 S(Z), where c > 0 is a absolute costat. Our first aim is to discuss similar iequalities for the surface area of hyperplae projectios of a arbitrary covex body K i R. I what follows, we deote by K the miimal surface area parameter of K, defied by (.4) K := mi {S(T (K))/ T (K) } : T GL(). It is kow that c K c for every covex body K i R, where c, c > 0 are absolute costats (see Sectio 2 for defiitios, refereces ad backgroud iformatio). Our aalogue of (.) is the followig theorem. Theorem.. There exists a absolute costat c > 0 such that, for every covex body K i R, (.5) K mi S(P ξ S ξ (K)) 2b K S(K) c K S(K). ω

2 This iequality is sharp e.g. for the Euclidea uit ball. Note that c K / c for every covex body K i R, ad hece we have the geeral upper boud (.6) K mi ξ S S(P ξ (K)) c S(K). Our method employs a estimate for the miimal volume of a hyperplae projectio of K: oe has (.7) mi P ξ S ξ (K) c K for a absolute costat c > 0. Assumig that K is i the miimal surface area positio we have a coverse of Theorem.: Theorem.2. Let K be a covex body i R which is i the miimal surface area positio. The, (.8) K mi S(P ξ (K)) c S(K), ξ S where c > 0 is a absolute costat. The estimate of Theorem.2 is sharp; we provide a example i which the two quatities i (.8) are of the same order, usig extremal (with respect to miimal hyperplae projectios) bodies of miimal surface area that were costructed i [2]. I the case where K is a projectio body, oe ca see that (.7) holds true with c replaced by b (see Sectio 3). This leads to a alterative proof of (.) with a weaker (by a factor of 2) costat. Theorem.3. Let Z be a projectio body i R. The, (.9) Z mi ξ S S(P ξ (Z)) 2b S(Z). It should be oted that there are covex bodies which are ot projectio bodies but their miimal surface area parameter K is of the order of ; a example is give by B, the uit ball of l. O the other had, there exist projectio bodies whose miimal surface area parameter is of the order of ; a example is give by the cube. Thus, the estimates of Theorem. ad Theorem.3 complemet each other. I our ext result we replace mi S(P ξ (K)) by the expectatio of S(P ξ (K)) o the sphere. Theorem.4. Let K be a covex body i R. The, (.0) K S(P ξ (K)) dσ(ξ) 2( )ω S 2 S(K) 2 c 2 S(K) 2, ω where c 2 > 0 is a absolute costat. A cosequece of Theorem.4 is that if K is i some of the classical positios (miimal surface area, isotropic or Joh s positio, or it is symmetric ad i Löwer s positio) the (.) K S(P ξ (K)) dσ(ξ) c S(K). S The reaso is that, i all these cases, the surface area of K satisfies a iequality of the form S(K) c K (see Sectio 2 for a brief descriptio of the classical positios of a covex body ad for a proof of this last assertio). Passig to lower bouds, our aalogue of (.3) is the followig theorem. 2

3 Theorem.5. Let K be a covex body i R. The, (.2) S(P ξ (K)) dσ(ξ) c 3 S(K) 2, S where c 3 > 0 is a absolute costat. A cosequece of Theorem.5 is that if K is i the miimal surface area, miimal mea width, isotropic, Joh or Löwer positio, the (.3) K S(P ξ (K)) dσ(ξ) c S(K), S where c > 0 is a absolute costat. I particular, (.4) K max ξ S S(P ξ (K)) c S(K). Note that (.4) is stroger tha (.3); moreover, for bouds of this type there is o eed to assume that K is a projectio body. I fact, our proof of Theorem.5 shows that (.3) cotiues to hold as log as (.5) S(K) c K for a absolute costat c > 0. This is a mild coditio which is satisfied ot oly by the classical positios but also by all reasoable positios of K. All these iequalities are proved i Sectio 4. Our mai tools are a result from [7] statig that (.6) S(P ξ (K)) P ξ (K) 2( ) S(K) K for every covex body K i R ad ay ξ S, estimates from [9] for the volume of the projectio body of a covex body i terms of its miimal surface area parameter, ad Aleksadrov s iequalities. For the class of projectio bodies, we prove ad use the followig sharp estimate (Lemma 3.): if Z is a projectio body i R the (.7) mi P ξ S ξ (Z) b Z. I Sectio 5 we study the same questios for the quermassitegrals V k (K) = V ((K, k), (B 2, k)) of a covex body K ad the correspodig quermassitegrals of its hyperplae projectios. We obtai the followig estimates: (i) For every p 2 we have (.8) K mi V p(p ξ (K)) (p + )ω K V p (K) c(p + ) K V p (K) ξ S ω ad (.9) K V p (P ξ (K)) dσ(ξ) (p + )ω S ω S(K) K V p (K). (ii) If Z is a projectio body i R the, for every p 2 we have (.20) Z mi ξ S V p(p ξ (Z)) (p + )b V p (Z). 3

4 (iii) If K is i the miimal surface area, isotropic or Joh s positio, or it is symmetric ad i Löwer s positio, the, for every p 2 we have (.2) K V p (P ξ (K)) dσ(ξ) c(p + ) V p (K). S (iv) For every p 2 we have (.22) V p (P ξ (K)) dσ(ξ) S ω p p ω [V p (K)] p p. (v) If K is i the miimal surface area, isotropic or Joh s positio, or it is symmetric ad i Löwer s positio the, for every p 2 we have (.23) K S V p (P ξ (K)) dσ(ξ) ω c ω p p 0 p p V p (K) ( c ) p 2( p) V p (K). The proofs employ the same tools as i the surface area case. The mai additioal igrediet is a geeralizatio of (.6) to subspaces of arbitrary dimesio ad quermassitegrals of ay order, proved i [5]: If K is a covex body i R ad 0 p k, the, for every F G,k, (.24) V p (K) K ( k+p k ) V k p(p F (K)). P F (K) This iequality allows us to obtai further geeralizatios of the results of Sectio 4; we ca compare the surface area of a covex body K to the miimal, average or maximal surface area of its lower dimesioal projectios P F (K), F G,k, for ay give k. This is doe i Sectio 6. There are several questios that arise from this work ad we hope that the reader might fid them iterestig; these are stated explicitely throughout the text. 2 Notatio ad backgroud We work i R, which is equipped with a Euclidea structure,. We deote by 2 the correspodig Euclidea orm, ad write B2 for the Euclidea uit ball ad S for the uit sphere. We deote the uit ball of l p by Bp, p. I particular, we also write Q for the cube B = [, ] ad C = [ 2, ] 2 for the cube of volume. Volume is deoted by. We write ω for the volume of B2 ad σ for the rotatioally ivariat probability measure o S. The Grassma maifold G,k of k-dimesioal subspaces of R is equipped with the Haar probability measure ν,k. For every k ad F G,k we write P F for the orthogoal projectio from R oto F, ad we set B F = B2 F ad S F = S F. Fially, we write A for the homothetic image of volume of a symmetric covex body A R, i.e. A := A A. The letters c, c, c, c 2 etc. deote absolute positive costats which may chage from lie to lie. Wheever we write a b, we mea that there exist absolute costats c, c 2 > 0 such that c a b c 2 a. Also, if K, L R we will write K L if there exist absolute costats c, c 2 > 0 such that c K L c 2 K. We refer to the books [6] ad [4] for basic facts from the Bru-Mikowski theory ad to the book [] for basic facts from asymptotic covex geometry. We also refer to [3] for more iformatio o isotropic covex bodies. 2.. Covex bodies. A covex body i R is a compact covex subset K of R with o-empty iterior. We say that K is symmetric if x K implies that x K, ad that K is cetered if its baryceter K K x dx is at the origi. The support fuctio of a covex body K is defied by h K (y) = max{ x, y : x K}, ad the mea width of K is (2.) w(k) = h K (θ) dσ(θ). S 4

5 The circumradius of K is the quatity R(K) = max{ x 2 : x K} i.e. the smallest R > 0 for which K RB2. If 0 it(k) the we write r(k) for the iradius of K (the largest r > 0 for which rb2 K) ad we defie the polar body K of K by (2.2) K := {y R : x, y for all x K}. The volume radius of K is the quatity vrad(k) = ( K / B 2 ) /. Itegratio i polar coordiates shows that if the origi is a iterior poit of K the the volume radius of K ca be expressed as ( ) / (2.3) vrad(k) = θ K dσ(θ), S where θ K = mi{t > 0 : θ tk}. We also defie (2.4) M(K) = θ K dσ(θ). S 2.2. Mixed volumes. From Mikowski s fudametal theorem we kow that if K,..., K m are o-empty, compact covex subsets of R, the the volume of t K + +t m K m is a homogeeous polyomial of degree i t i > 0. That is, (2.5) t K + + t m K m = V (K i,..., K i )t i t i, i,...,i m where the coefficiets V (K i,..., K i ) are chose to be ivariat uder permutatios of their argumets. The coefficiet V (K,..., K ) is the mixed volume of K,..., K. I particular, if K ad C are two covex bodies i R the the fuctio K + tc is a polyomial i t [0, ): (2.6) K + tc = j=0 ( ) V j (K, C) t j, j where V j (K, C) = V ((K, j), (C, j)) is the j-th mixed volume of K ad C (we use the otatio (C, j) for C,..., C j-times). If C = B2 the we set V j (K) := V j (K, B2 ) = V ((K, j), (B2, j)); this is the j-th quermassitegral of K. Note that (2.7) V (K, C) = lim K + tc K, t 0 + t ad by the Bru-Mikowski iequality we see that (2.8) V (K, C) K C / for all K ad C (this is Mikowski s first iequality). The mixed volume V (K, C) ca be expressed as (2.9) V (K, C) = h C (θ)dσ K (θ), S where σ K is the surface area measure of K. I particular, the surface area of K satisfies (2.0) S(K) = V (K). We will also use the Aleksadrov iequalities: if K is a covex body i R the the sequece ( (2.) Q k (K) = P F (K) dν,k (F ) ω k G,k 5 ) k

6 is decreasig i k. This is a cosequece of the Aleksadrov-Fechel iequality (see [4] ad [4]). I particular, for every k we have (2.2) ( K ω ) ( ) k P F (K) dν,k (F ) ω k G,k w(k) Classical positios. Let K be a cetered covex body i R. We itroduce the classical positios of K that we are goig to discuss; we set the otatio ad provide some backgroud iformatio. Miimal surface area positio. We say that K has miimal surface area if S(K) S(T (K)) for every T SL(). Recall that the area measure σ K of K is the Borel measure o S defied by (2.3) σ K (A) = λ({x bd(k) : the outer ormal to K at x belogs to A}), where λ is the usual surface measure o K. Petty ([3], see also [9]) proved that K has miimal surface area if ad oly if σ K satisfies the isotropic coditio (2.4) S(K) = ξ, θ 2 dσ K (θ) S for every ξ S. From the isoperimetric iequality we kow that S(K) ω K. The reverse isoperimetric iequality of K. Ball [2] states that if K has miimal surface area ad volume the S(K) S(C ) = 2 i the symmetric case ad S(K) S( ) c 0 i the ot ecessarily symmetric case, where is a regular simplex of volume i R ad c 0 > 0 is a absolute costat. Miimal mea width positio. We say that K is i miimal mea width positio if w(k) w(t (K)) for every T SL(). It was proved i [8] that K has miimal mea width if ad oly if (2.5) w(k) = ξ, θ 2 h K (θ)dσ(θ) S for every ξ S. From results of Figiel-Tomczak, Lewis ad Pisier (see [, Chapter 6]) it follows that if a symmetric covex body K i R has miimal mea width the (2.6) M(K)w(K) c log(d K + ) where d K := d(k, B2 ) is the Baach-Mazur distace from K to B2 ad c > 0 is a absolute costat. If we assume that K = the it is easy to check that M(K) c/, ad hece from (2.6) we see that w(k) c log(d K + ). The, a simple argumet shows that a ot ecessarily symmetric covex body of volume i R that has miimal mea width satisfies a similar boud: w(k) c log. Isotropic positio. For every cetered covex body K of volume i R ad ay q we defie ( (2.7) I q (K) = K x q 2 dx ) /q. We say that K is i the isotropic positio if I 2 (K) I 2 (T (K)) for every T SL(). This is equivalet to the existece of a costat L K > 0 such that (2.8) x, ξ 2 dx = L 2 K for every ξ S. It is kow that if K is cetered the (2.9) x, ξ 2 dx K ξ 2 K K 6

7 for every ξ S. Therefore, if K is isotropic we see that all hyperplae sectios K ξ of K have volume equal (up to a absolute costat) to L K. Joh ad Löwer positio. We say that a covex body K is i Joh s positio if the ellipsoid of maximal volume iscribed i K is a multiple of the Euclidea uit ball B2. We say that a covex body K is i Löwer s positio if the ellipsoid of miimal volume cotaiig K is a multiple of the Euclidea uit ball B2. Oe ca check that this holds true if ad oly if K is i Joh s positio. The volume ratio of a cetered covex body K i R is the quatity { ( K ) } (2.20) vr(k) = if : E is a ellipsoid ad E K. E The outer volume ratio of a cetered covex body K i R is the quatity ovr(k) = vr(k ). K. Ball proved i [2] that if K is i Joh s positio the vr(k) vr(c ) i the symmetric case ad vr(k) vr( ) i the ot ecessarily symmetric case; i fact, the reverse isoperimetric iequality follows from this fact Surface area ad iradius. Let K be a cetered covex body i R. Recall that the iradius r(k) of K is the largest r > 0 for which rb 2 K. Usig the mootoicity of mixed volumes we may write (2.2) S(K) = V (K, B 2 ) V (K, ) r(k) K. Sice the mixed volumes are homogeeous with respect to each of their argumets ad V (K,..., K) = K, we have the followig geeral estimate for the surface area S(K) of K. Lemma 2.. Let K be a covex body i R with 0 it(k). The, (2.22) S(K) K r(k). Usig Lemma 2. we obtai upper bouds for the surface area of a body which is i isotropic, Joh s or Löwer s positio. Propositio 2.2. Let K be a cetered covex body of volume i R. (i) If K is isotropic the S(K) c/l K c, where c, c > 0 are absolute costats. (ii) If K is i miimal surface area positio or i Joh s positio the S(K) c, where c > 0 is a absolute costat. (iii) If K is symmetric ad i Löwer s positio the S(K) c, where c > 0 is a absolute costat. (iv) If K is symmetric ad i the miimal mea width positio the S(K) c log, where c > 0 is a absolute costat. Proof. The iclusio L K B 2 K for a isotropic symmetric covex body K i R is clear sice h K (u) =, u L (K), u L2 (K) = L K for every u S. This shows that r(k) L K i this case. If K is cetered but ot ecessarily symmetric, the we still have h K (u) cl K : to see this, we use the fact that e max{ K (tθ +θ ) : t R} K θ (see [3, Chapter 2]) ad the write L K h K(u) c h K (u) K θ c 2 K (tθ + θ ) dt = c 2 {x K : x, θ 0} c 3, 0 7

8 where c 3 > 0 is a absolute costat (the last iequality follows from Grübaum s lemma, see [3, Chapter 2]). To coclude the proof of (i) we recall that L K c for ay covex body K i R. Assume that K is i Joh s positio. The, usig the volume ratio estimate we see that (2.23) ( ) K r(k) = vr(k) c, r(k)b2 which implies that r(k) c, ad hece S(K) c. It follows that if K is i miimal surface area positio we also have S(K) c. Next, assume that K is symmetric ad i Löwer s positio; this time we use the fact that R(K) r(k) by Joh s theorem, ad the (2.24) = K / R(K)B 2 / cr(k)/ cr(k). Fially, if K is symmetric ad i the miimal mea width positio we ca use the direct estimate (2.25) R(K ) c w(k ) = c M(K) c log which is a cosequece of (2.6) ad of the fact that w(k) c by Urysoh s iequality. This shows that r(k) = /R(K ) c/ log, ad (iv) follows. Note. The example of the cube C shows that the bouds (i), (ii) ad (iii) of Propositio 2.2 are sharp up to a absolute costat. 3 Projectios of projectio bodies A zooid is the limit of Mikowski sums of lie segmets i the Hausdorff metric. Equivaletly, a symmetric covex body Z is a zooid if ad oly if its polar body is the uit ball of a -dimesioal subspace of a L -space; i.e. if there exists a positive measure µ (the supportig measure of Z) o S such that (3.) h Z (x) = x Z = x, y dµ(y). 2 S The class of origi-symmetric zooids coicides with the class of projectio bodies. Recall that the projectio body ΠK of a covex body K is the symmetric covex body whose support fuctio is defied by (3.2) h ΠK (ξ) = P ξ (K), ξ S. From Cauchy s formula (3.3) P ξ (K) = 2 S ξ, θ dσ K (θ), where σ K is the surface area measure of K, we see that the projectio body of K is a zooid whose supportig measure is σ K. Mikowski s existece theorem implies that, coversely, every origi-symmetric zooid is the projectio body of some symmetric covex body i R. Moreover, if we deote by C the class of origi-symmetric covex bodies ad by Z the class of origi-symmetric zooids, Aleksadrov s uiqueess theorem shows that the Mikowski map Π : C Z with K ΠK, is ijective. Note also that Z is ivariat uder ivertible liear trasformatios ad closed i the Hausdorff metric. Let K be a covex body of volume i R. The, the volume of ΠK ad of its polar body Π K satisfy the bouds (see [9]) (3.4) ( K ) ΠK ω ( ω K ω ) 8

9 ad (3.5) ω ( ω ω K ) Π K 4! K. All these iequalities are sharp as oe ca see from the examples of the ball ad the cube. Our ext lemma provides a estimate for the volume of the miimal hyperplae projectio of a zooid. Lemma 3.. Let Z be a zooid i R. The, (3.6) mi P ξ (Z) b Z. ξ S Proof. We write Z = ΠK for some covex body K. Recall the volume formula for zooids (3.7) Z = V (Z, ΠK) = V (K, ΠZ) = h ΠZ (ξ) dσ K (ξ) = P S ξ (Z) dσ K (ξ). S Therefore, (3.8) Z S(K) O the other had, by (3.4) we have mi ξ S P ξ (Z). (3.9) Z = ΠK ω K ω K b S(K), where we used the defiitio (.2) of the costat b ad the fact that S(K) K K (.4) of the miimal surface area parameter K. The, by the defiitio (3.0) S(K) mi P ξ (Z) Z Z b S(K) ξ S Z, ad the result follows. Sice every projectio of a zooid is a zooid, a simple iductio argumet leads to the followig geeral result. Theorem 3.2. Let Z be a zooid i R. The, for every k we have (3.) mi F G,k P F (Z) ϱ,k Z k, where (3.2) ϱ,k = s=k+ sb s s = ω s s=k+ ω s s s Questio 3.3. The example of the ball shows that Lemma 3. is sharp. It would be iterestig to establish the precise costat i Theorem

10 4 Surface area of hyperplae projectios Our geeralizatio of (.) is i terms of the miimal surface area parameter K of K. Theorem 4.. Let K be a covex body i R. The, (4.) K mi S(P ξ S ξ (K)) 2b K S(K) c K S(K), ω where c > 0 is a absolute costat ad K geeral we have that is the miimal surface area parameter of K. Therefore, i (4.2) K mi ξ S S(P ξ (K)) c S(K). The mai igrediet i the proof is the ext result from [7]. Lemma 4.2. If K is a covex body i R the (4.3) S(P ξ (K)) P ξ (K) 2( ) S(K) K for every ξ S. Proof of Theorem 4.. From (4.3) we have (4.4) K S(P ξ (K)) for every ξ S. Therefore, 2( ) S(K) P ξ (K) (4.5) K mi S(P 2( ) ξ S ξ (K)) S(K) mi P ξ S ξ (K). Next, we observe that (4.6) mi P ξ S ξ (K) = mi h ΠK(ξ) = r(πk). ξ S Sice (4.7) ΠK ω K we get ω K (4.8) r(πk) vrad(πk) ω K ω Goig back to (4.5) we see that K. (4.9) K mi S(P ξ (K)) 2( )ω K ξ S 2 S(K) K, ω ad this proves (4.). 0

11 Questio 4.3. It would be iterestig to decide whether there exist covex bodies K such that (4.0) K mi ξ S S(P ξ (K)) c S(K). This would show that Theorem 4. is asymptotically sharp. Note that i the case of the Euclidea ball oe has (4.) B 2 mi ξ S S(P ξ (B 2 )) c S(B 2 ). We ca prove a iequality which is reverse to (4.) for ay covex body K which is i the miimal surface area positio, usig the followig fact (see Theorem.2 i [7]): for ay covex body K i R ad ay ξ S oe has (4.2) V 2 (K) 2V (K) V 2(P ξ (K)). P ξ (K) Note that (4.3) V 2 (K) = ω [Q 2 (K)] 2 ω [Q (K)] 2 = ω while (4.4) V (K) = S(K), ( )V 2 (P ξ (K)) = S(P ξ (K)) ad if K is i the miimal surface area positio by (2.4) we also have (4.5) P ξ (K) = u, ξ dσ K (u) u, ξ 2 dσ K (u) = S(K) 2 S 2 S 2 Combiig the above we get (4.6) S(P ξ (K)) Therefore, we have (4.7) K mi ξ S S(P ξ (K)) Sice S(K) = K K, we get: ( )ω 4 ( )ω [V (K)] 2. 4 [V (K)] 2 K [V (K)] 2, ( )ω = 4 2 = V (K). 2 [S(K)] 2 K. (4.8) K mi ξ S S(P ξ (K)) This proves the followig: 4 2 K ( )ω S(K). Theorem 4.4. Let K be a covex body i R which is i the miimal surface area positio. The, (4.9) K mi S(P ξ S ξ (K)) c S(K), where c > 0 is a absolute costat.

12 Remark 4.5. Theorem 4.4 is sharp. I [2] it is proved that there exists a ucoditioal covex body K 0 of volume i R which has miimal surface area ad satisfies (4.20) mi P ξ S ξ (K 0) c, where c > 0 is a absolute costat. From (4.5) we see that 2( ) (4.2) K 0 mi S(P ξ (K 0 )) S(K 0 ) mi P ξ S ξ (K 0 ) 2c S(K 0 ), ξ S ad sice K 0 = we get: Propositio 4.6. There exists a ucoditioal covex body K 0 i R which has miimal surface area ad satisfies (4.22) K 0 mi S(P ξ (K 0 )) c S(K 0 ), ξ S where c > 0 is a absolute costat. Next, assume that K = Z is a zooid. Repeatig the previous argumet, but usig ow Lemma 3. i order to estimate r(πz), we obtai a alterative proof of (.) (with a weaker, by a factor of 2, costat). Theorem 4.7. Let Z be a zooid i R. The, (4.23) Z mi ξ S S(P ξ (Z)) 2b S(Z). Proof. We have 2( ) (4.24) Z mi S(P ξ (Z)) S(Z) ξ S 2( ) mi P ξ (Z) ξ S b S(Z) Z by Lemma 3.. Dividig by Z we get the result. Questio 4.8. It would be iterestig to decide whether i the case of zooids oe has (4.25) Z S(P ξ (Z)) for every ξ S. Theorem 4.7. ( ) S(Z) P ξ (Z) This improvemet of (4.3) (for the class of zooids) would give a sharp versio of Next, we pass to estimates for the average surface area of hyperplae projectios of K. Theorem 4.9. Let K be a covex body i R. The, (4.26) K S(P ξ (K)) dσ(ξ) 2b S(K) 2 c S(K) 2, S ω where c > 0 is a absolute costat. Proof. From (4.3) we have (4.27) K S(P ξ (K)) 2( ) S(K) P ξ (K) 2

13 for every ξ S. Itegratig o S ad usig the idetity (4.28) S(K) = ω P ω ξ (K) dσ(ξ) S we get (4.29) Sice (4.30) we get the result. 2( ) K S(P ξ (K)) dσ(ξ) S(K) P S ξ (K) dσ(ξ) S 2( ) ω = S(K) 2 ω 2( ) ω = 2b, ω ω Now, let us assume that K is i the miimal surface area, isotropic or Joh s positio, or it is symmetric ad i Löwer s positio. The, from Propositio 2.2 we kow that (4.3) S(K) c 0 K, where c 0 > 0 is a absolute costat. From Theorem 4.9 we get: Theorem 4.0. Let K be a covex body i R. If K is i the miimal surface area, isotropic or Joh s positio, or it is symmetric ad i Löwer s positio, the (4.32) K S(P ξ (K)) dσ(ξ) c 2 S(K) S where c 2 > 0 is a absolute costat. Note. If K is symmetric ad i the miimal mea width positio, usig Propositio 2.2 agai, we get a weaker (by a log term) result: (4.33) K S(P ξ (K)) dσ(ξ) c 2 (log ) S(K) S where c 2 > 0 is a absolute costat. We pass ow to lower bouds. Our aalogue of (.3) is the ext theorem. Theorem 4.. Let K be a covex body i R. The, (4.34) S(P ξ (K)) dσ(ξ) c 3 S(K) 2, S where c 3 > 0 is a absolute costat. Proof. We write S(P ξ (K)) dσ(ξ) = ( )ω (4.35) P S ω ξ,θ (K) dσ ξ (θ) dσ(ξ) 2 S S ξ = ( )ω P F (K) dν, 2 (F ). ω 2 G, 2 3

14 From the Aleksadrov iequalities it follows that (4.36) ( ω 2 G, 2 P F (K) dν, 2 (F ) ) 2 ( ω ( S(K) = ω P ξ (K) dσ(ξ) S ), ) which gives (4.37) S(P ξ (K)) dσ(ξ) ( )ω S (ω ) 2 S(K) 2 c3 S(K) 2, where c 3 > 0 is a absolute costat. Now, let us assume that K is i the miimal surface area, miimal mea width, isotropic, Joh or Löwer positio. The, from Propositio 2.2 (or from simple estimates i the cases of a ot ecessarily covex body K that are ot covered there) we kow that, e.g. (4.38) S(K) c 0 2 K, where c 0 > 0 is a absolute costat. It follows that (4.39) S(K) (c0 2 ) 2 K c4 K, where c 4 > 0 is a absolute costat. The, (4.40) K S(K) 2 c 4 S(K). Thus we have proved: Theorem 4.2. Let K be a covex body i R. If K is i the miimal surface area, miimal mea width, isotropic, Joh s or Löwer s positio, the (4.4) K S(P ξ (K)) dσ(ξ) c 5 S(K), S where c 5 > 0 is a absolute costat. Note. The proof of Theorem 4.2 shows that (4.4) cotiues to hold as log as the mild coditio (4.42) S(K) c K is satisfied by K with a absolute costat c > 0. 5 Quermassitegrals of hyperplae projectios A geeralizatio of (4.3) to subspaces of arbitrary dimesio ad quermassitegrals of ay order was give i [5]. Theorem 5.. Let K be a covex body i R ad let 0 p k. The for every k-dimesioal subspace F of R, if P F (K) deotes the orthogoal projectio of K oto F, we have (5.) V p (K) K ( k+p k ) V k p(p F (K)). P F (K) 4

15 Settig k =, for every p 2 we have (5.2) V p (K) K p + V p (P ξ (K)). P ξ (K) Therefore, (5.3) K mi V p(p ξ (K)) (p + ) V p (K) mi P ξ S ξ S ξ (K), ad usig (4.8) ad Lemma 3. we immediately get the followig theorem. Theorem 5.2. Let K be a covex body i R. For every p 2 we have (5.4) K mi V p (P ξ (K)) (p + )ω K V p (K) c (p + ) K V p (K), ξ S ω where c > 0 is a absolute costat. If Z is a zooid i R the, for every p 2 we have (5.5) Z mi ξ S V p(p ξ (Z)) (p + )b V p (Z). Startig from (5.2) ad itegratig o the sphere we get (5.6) K V p (P ξ (K)) dσ(ξ) (p + )V p (K) S P ξ (K) dσ(ξ) S = (p + )ω V p (K)S(K). ω Dividig by K we get: Theorem 5.3. Let K be a covex body i R. For every p 2 we have (5.7) K V p (P ξ (K)) dσ(ξ) (p + )ω S(K) V p (K). S ω K I particular, if K is i the miimal surface area, isotropic or Joh s positio, or it is symmetric ad i Löwer s positio, the we have (5.8) K V p (P ξ (K)) dσ(ξ) c (p + ) V p (K), S where c > 0 is a absolute costat. For the lower boud, a aalogue of Theorem 4., we first observe that (5.9) V p (K) = ω [Q p (K)] p ad V p (P ξ (K)) = ω [Q p (P ξ (K))] p for every ξ S. The, we write (5.0) V p (P ξ (K)) dσ(ξ) = S ω ω p = ω ω p S G ξ (, p) G, p P F (K) dν, p (F ) = ω [Q p (K)] p. P E (K) d(e) dσ(ξ) 5

16 From the Aleksadrov iequalities we have Q p (K) Q p (K), ad hece (5.) V p (P ξ (K)) dσ(ξ) ω [Q p (K)] 2 p S which gives the ext theorem. ( ) p V p p (K) ω ω Theorem 5.4. Let K be a covex body i R. For every p 2 we have (5.2) V p (P ξ (K)) dσ(ξ) ω [V p (K)] p p. S Usig the mootoicity of mixed volumes we may write p p ω (5.3) V p (K) V p ( (K, p), (r(k) K, p) ) K r(k) p. Now, let us assume that K is i the miimal surface area, isotropic or Joh s positio, or it is symmetric ad i Löwer s positio. The, r(k) c 0 K for a absolute costat c 0 > 0, ad (5.3) gives (5.4) K p c p 0 V p(k). Therefore, (5.5) K [V p (K)] p p From Theorem 5.4 we get: p p c0 [V p (K)] p [V p (K)] p p p p = c0 V p (K). Theorem 5.5. Let K be a covex body i R, which is i the miimal surface area, isotropic or Joh s positio, or it is symmetric ad i Löwer s positio. For every p 2 we have (5.6) K where c > 0 is a absolute costat. Note that ( c ) S V p (P ξ (K)) dσ(ξ) ω c p 2( p) ω p p 0 p p V p (K) ( c ) p 2( p) c 2 for a absolute costat c 2 > 0 as log as p c/(log ). 6 Surface area of projectios of higher codimesio V p (K), Recall that mv m (A) = S(A) for every covex body A i R m. Therefore, settig p = i Theorem 5. we get: Lemma 6.. Let K be a covex body i R ad let k. The for every k-dimesioal subspace F of R we have (6.) S(K) K k( k + ) S(P F (K)). P F (K) We first prove a aalogue of Theorem

17 Theorem 6.2. Let Z be a zooid i R ad let k. The, (6.2) Z k where ϱ,k is the costat i Theorem 3.2. Proof. From (6.) we see that k( k + ) mi S(P F (Z)) ϱ,k S(Z), F G,k k( k + ) k( k + ) (6.3) Z mi S(P F (Z)) S(Z) mi P F (Z) ϱ,k S(Z) Z k, F G,k F G,k where i the last step we have also used Theorem 3.2. Dividig by Z k we get the result. Defiitio 6.3. For every covex body K i R ad every k we itroduce the parameter (6.4) p k (K) := P K k F (K) dν,k (F ). G,k Usig Lemma 6. ad applyig the same argumet as i the proof of Theorem 4.9 we get: Theorem 6.4. Let K be a covex body i R ad let k. The, (6.5) K k k( k + ) S(P E (K)) dν,k (E) S(K) p k (K). G,k Proof. From Lemma 6. we have (6.6) K S(P F (K)) k( k + ) for every F G,k. Itegratig with respect to ν,k o G,k we get (6.7) K S(P F (K)) dν,k (F ) G,k k( k + ) S(K) P F (K) S(K) P F (K) dν,k (F ), G,k ad the result follows. Remark 6.5. Let us assume that K is i the miimal surface area, isotropic or Joh s positio, or it is symmetric ad i Löwer s positio. From (5.3) ad the fact that r(k) c 0 K we get (6.8) p k (K) = K k ω k V k (K) ω k ω K k ω The, Theorem 6.4 gives the followig aalogue of Theorem 4.0: K c k 0 K k = ω k ω c k. 0 Theorem 6.6. Let K be a covex body i R. If K is i the miimal surface area, isotropic or Joh s positio, or it is symmetric ad i Löwer s positio, the (6.9) K k k( k + ) ω k S(P F (K)) dν,k (F ) G,k ω c k S(K) 0 where c 0 > 0 is a absolute costat. The lower boud of Theorem 4. ca be geeralized as follows. 7

18 Theorem 6.7. Let K be a covex body i R. For every k we have kω k (6.0) S(P F (K)) dν,k (F ) S(K) k G,k (ω ) k. Proof. We write S(P F (K)) dν,k (F ) = kω k (6.) P G,k ω F ξ (K) dσ F (ξ) dν,k (F ) k G,k E ξ = kω k P E (K) dν,k (E). ω k G,k From the Aleksadrov iequalities we have (6.2) ( ω k G,k P E (K) dν,k (E) ) k ( ω ( S(K) = ω P ξ (K) dσ(ξ) S ), ) which gives (6.3) G,k S(P F (K)) dν,k (F ) kω k (ω ) k S(K) k, as claimed. Now, let us assume that K is i the miimal surface area, isotropic or Joh s positio, or it is symmetric ad i Löwer s positio. The, from Propositio 2.2 we kow that (6.4) S(K) c 0 K, where c 0 > 0 is a absolute costat. Therefore, (6.5) K k k S(K) S(K) k k (c 0 ) k S(K) ad Theorem 6.7 implies the followig. = (c 0 ) k S(K), Theorem 6.8. Let K be a covex body i R. If K is i the miimal surface area, isotropic or Joh s positio, or it is symmetric ad i Löwer s positio, the (6.6) K k kω k S(P F (K)) dν,k (F ) S(K), G,k (ω ) k (c0 ) k where c 0 > 0 is a absolute costat. Ackowledgemet. The secod amed author was partially supported by the US Natioal Sciece Foudatio grat DMS

19 Refereces [] S. Artstei-Avida, A. Giaopoulos ad V. D. Milma, Asymptotic Geometric Aalysis, Vol. I, Mathematical Surveys ad Moographs 202, Amer. Math. Society (205). [2] K. M. Ball, Volume ratios ad a reverse isoperimetric iequality, J. Lodo Math. Soc. (2) 44 (99), [3] S. Brazitikos, A. Giaopoulos, P. Valettas ad B-H. Vritsiou, Geometry of isotropic covex bodies, Mathematical Surveys ad Moographs 96, Amer. Math. Society (204). [4] Y. D. Burago ad V. A. Zalgaller, Geometric Iequalities, Spriger Series i Soviet Mathematics, Spriger- Verlag, Berli-New York (988). [5] M. Fradelizi, A. Giaopoulos ad M. Meyer, Some iequalities about mixed volumes, Israel J. Math. 35 (2003), [6] R. J. Garder, Geometric Tomography, Secod Editio Ecyclopedia of Mathematics ad its Applicatios 58, Cambridge Uiversity Press, Cambridge (2006). [7] A. Giaopoulos, M. Hartzoulaki ad G. Paouris, O a local versio of the Aleksadrov-Fechel iequality for the quermassitegrals of a covex body, Proc. Amer. Math. Soc. 30 (2002), [8] A. Giaopoulos ad V. D. Milma, Extremal problems ad isotropic positios of covex bodies, Israel J. Math. 7 (2000), [9] A. Giaopoulos ad M. Papadimitrakis, Isotropic surface area measures, Mathematika 46 (999), -3. [0] A. Koldobsky, Stability ad separatio i volume compariso problems, Math. Model. Nat. Pheom. 8 (203), [] A. Koldobsky, Stability iequalities for projectios of covex bodies, Preprit; arxiv: [2] E. Markessiis, G. Paouris ad Ch. Saroglou, Comparig the M-positio with some classical positios of covex bodies, Math. Proc. Cambridge Philos. Soc. 52 (202), [3] C. M. Petty, Surface area of a covex body uder affie trasformatios, Proc. Amer. Math. Soc. 2 (96), [4] R. Scheider, Covex Bodies: The Bru-Mikowski Theory, Secod expaded editio. Ecyclopedia of Mathematics ad Its Applicatios 5, Cambridge Uiversity Press, Cambridge, 204. Apostolos Giaopoulos: Departmet of Mathematics, Uiversity of Athes, Paepistimioupolis 57-84, Athes, Greece. apgiaop@math.uoa.gr Alexader Koldobsky: Departmet of Mathematics, Uiversity of Missouri, Columbia, MO koldobskiya@missouri.edu Petros Valettas: Departmet of Mathematics, Uiversity of Missouri, Columbia, MO valettasp@missouri.edu 9