The Petty projection inequality for sets of finite perimeter
aa r X i v : . [ m a t h . F A ] F e b Noname manuscript No. (will be inserted by the editor)
The Petty projection inequality for sets of finite perimeter
Youjiang Lin the date of receipt and acceptance should be inserted later
Abstract
The Petty projection inequality for sets of finite perimeter is proved. Our approach is based onSteiner symmetrization. Neither the affine Sobolev inequality nor the functional Minkowski problem isused in our proof. Moreover, for sets of finite perimeter, we prove the Petty projection inequality withrespect to Steiner symmetrization.
Keywords
Petty projection inequality · set of finite perimeter · Steiner symmetrization
Mathematics Subject Classification (2010)
Within the Brunn-Minkowski theory, the two classical inequalities which connect the volume of aconvex body with that of its polar projection body are the Petty and Zhang projection inequalities. Unlikethe classical isoperimetric inequality (see, e.g., [19–21, 52]), the Petty and Zhang projection inequalitiesare affine isoperimetric inequalities in that they are inequalities between a pair of geometric functionalswhose product is invariant under affine transformations. Many important results about affine isoperimetricinequalities and their functional forms have been found (see, e.g., [3, 7, 10, 17, 28–31, 43]). The Pettyprojection inequality strengthens and directly implies the classical isoperimetric inequality, but it canbe viewed as an optimal isoperimetric inequality. It is the geometric core of the affine Sobolev-Zhanginequality [61] which strengthens the classical sharp Euclidean Sobolev inequality. The L p version ofPetty’s inequality by Lutwak et al. [43] and its Orlicz extension by the same authors [48] both representlandmark results in the evolution of the Brunn-Minkowski theory first towards an L p theory and, morerecently, towards an Orlicz theory of convex bodies. The L p and Orlicz theories of convex bodies haveexpanded rapidly (see, e.g., [7, 12, 22, 23, 26, 27, 32, 34–42, 44–47, 60, 62]).Affine isoperimetric inequalities referring to convex bodies have been extended to certain classes ofnon-convex domains such as star bodies and sets of finite perimeter. The Petty projection inequality has Y. LinSchool of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, PR ChinaE-mail: [email protected] Research of the author is supported by National Natural Science Foundation of China NSFC 11971080 and Natural ScienceFoundation Projection of Chongqing cstc2018jcyjAX0790. Y. Lin been generalized first to compact domains with smooth boundary by Zhang [61] and to sets of finiteperimeter by Wang [58]. Recently, Haberl and Schuster [30] show every even, zonal measure on theEuclidean unit sphere gives rise to an isoperimetric inequality for sets of finite perimeter which directlyimplies the classical Euclidean isoperimetric inequality. The strongest member of this large family ofinequalities was shown to be the only affine invariant one among them–the Petty projection inequality.In [58], Wang used the functional Minkowski problem on BV ( R n ) and an approach to the affine Sobolev-Zhang inequality for compactly supported C functions developed by Lutwak et al. [46] to establish theaffine Sobolev-Zhang inequality of BV functions. As a consequence, the Petty projection inequality forsets of finite perimeter was established. Using the same ideas, Haberl and Schuster [30] obtained the largefamily of sharp Sobolev type inequalities which can be seen as the functional form of the large family ofisoperimetric inequalities.In this paper, we give a direct proof of the Petty projection inequality for sets of finite perimeter.Our approach is based on Steiner symmetrization. Neither the affine Sobolev inequality nor the func-tional Minkowski problem is used in our proof. Moreover, for sets of finite perimeter, we prove the Pettyprojection inequality with respect to Steiner symmetrization.Let C n denote the set of compact sets of finite perimeter in R n with nonempty interior. For E ∈ C n ,let ∂ ∗ E and ν E = ( ν E , . . . , ν En − , ν Ey ) denote the reduced boundary and the generalized inner normal to E ,respectively (see Section 3.2 for precise definitions). We define the projection body Π E of E to be theconvex set with support function h ( Π E , z ) = Z ∂ ∗ E | z · ν E ( x ) | d H n − ( x ) , z ∈ R n . (1.1)This extension of Minkowski’s classical notion of the projection body of a convex body was first given byWang [58], who generalized a definition of Zhang [61] for compact sets with piecewise C boundary.For the polar (see Section 2.1 for definitions) of Π E we will write Π ∗ E . The L n Lebesgue measure ofa set of finite perimeter E ∈ C n will be denoted by | E | , and for the volume of the unit ball in R n we use ω n . We assume that E ∈ C n satisfies the condition H n − ( { x ∈ ∂ ∗ E : ν Ey ( x ) = } ) = . (1.2) Theorem 1.1
Let E ∈ C n satisfy (1.2) and E s denote its Steiner symmetrization. Then | E | n − | Π ∗ E | ≤ | E s | n − | Π ∗ E s | . (1.3) Theorem 1.2
Let E ∈ C n . Then | E | n − | Π ∗ E | ≤ ( ω n / ω n − ) n . (1.4)Note that since the absolute value function | · | is not strictly convex, we cannot characterize the casesof equality in (1.3) and (1.4) using the methods in the paper.In section 2, we set up notation and terminology and compile some basic facts about the Brunn-Minkowski theory of convex bodies and the theory of functions of bounded variation. In Section 3, weprove that for any a set of finite perimeter E , there exists a sequence of unit directions { u i } such that thereexists a subsequence of the successive Steiner symmetrizations E i : = S u i · · · S u E of E which convergesto an origin-symmetric ball with same volume. The main difficulty in proving the convergence is thatevery direction u i of Steiner symmetrization is related to E i − . Section 4 is devoted to the study of somebasic properties of projection bodies of sets of finite perimeter. For example, the continuity and the affine he Petty projection inequality for sets of finite perimeter 3 invariance of the Petty projection operator. It is worth pointing out that the proofs of these properties arenew and did not appear in the known literatures.In Section 5, we give the proofs of main theorems. First, we prove the monotonicity of the volumesof projection bodies with respect to Steiner symmetrization. To prove the monotonicity, we make criticaluse of a beautiful fact found by Lutwak et al. in [48]. The fact gives a useful way to imply the inclusionrelation between the Steiner symmetrization of a convex body and another convex body. In the remarkablepaper [13], Chleb´ık et al. characterized the sets of finite perimeter whose perimeter is preserved underSteiner symmetrization and proved the monotonicity of the perimeter of sets of finite perimeter withrespect to Steiner symmetrization. The ideas and techniques of Chleb´ık et al. [13] play a critical role inthis paper. Comparing to the paper [13], the perimeter of sets of finite perimeter is replaced by the volumeof the projection body of sets of finite perimeter, which is a function of the volume of the unit ball of an n -dimensional Banach space. Finally, using the convergence of Steiner symmetrization, monotonicity andcontinuity of the projection operator, we prove the Petty projection inequality for sets of finite perimeter. Let N denote the set of positive integers. Let { e ,. . . , e n } denote the standard orthonormal basis of theEuclidean space R n . A point x ∈ R n , n ≥
2, will be usually labeled by ( x ′ , y ) , where x ′ = ( x , . . . , x n − ) ∈ R n − and y ∈ R . To emphasize the different role of the variable y , we shall also write R n = R n − × R y .We shall use R + and R − to denote [ , + ∞ ) and ( − ∞ , ] , respectively. For u ∈ S n − , let u ⊥ denote thecodimension 1 subspace of R n that is orthogonal to u . If E is a measurable subset of R n and E is containedin an i -dimensional affine subspace of R n but in no affine subspace of lower dimension, then | E | will denotethe i -dimensional Lebesgue measure of E . Two measurable sets E and F are equivalent if the Lebesguemeasure of their symmetric difference E △ F is zero. Let B n denote the Euclidean unit ball centered at theorigin in R n . Let B r ( x ) denote the ball, centered at x , having radius r . If x ∈ R n then by abuse of notationwe will write | x | = √ x · x .2.1 Convex bodies.We develop some notation and, for quick later reference, list some basic facts about convex bodies.Good general references for the theory of convex bodies are provided by the books of Gardner [23],Gruber [24, 25], Webster [59] and Schneider [53].We write K n for the set of convex bodies (compact convex subsets) of R n . We write K n for the set ofconvex bodies that contain the origin in their interiors. For K ∈ K n , let h ( K ; · ) = h K : R n → R denote the support function of K ; i.e., h ( K ; x ) = max { x · z : z ∈ K } . For K ∈ K n , its polar body K ∗ is defined by K ∗ = { x ∈ R n : x · z ≤ z ∈ K } . For A ∈ SL ( n ) and K ∈ K n , ( AK ) ∗ = A − t K ∗ . (2.1)For K ∈ K n , its radial function ρ K is defined as ρ K ( x ) = max { λ ≥ λ x ∈ K } , x ∈ R n \{ } . (2.2) Y. Lin
It is easily verified that ρ K ∗ ( x ) = h K ( x ) , x ∈ R n \ { } . (2.3)2.2 Functions of bounded variation and Sets of finite perimeterIn this section, we review some basic definitions and facts about functions of bounded variation andsets of finite perimeter on R n . Good general references are Ambrosio et al. [1], Cianchi [14], Evans [18],Maz ′ ya [50], [51] and Ziemer [63].The space of functions of bounded variation in R n is denoted by BV ( R n ) . Recall that a function f ∈ L ( R n ) is said to be of bounded variation in R n if it is integrable and its distributional gradient D f isa vector-valued Radon measure in R n whose total variation | D f | is finite in R n ; thus BV ( R n ) = L ( R n ) ∩ { f : D f is a measure , | D f | ( R n ) < ∞ } . For f ∈ L ( R n ) , let f ∗ denote the precise representative of f , i.e. f ∗ ( x ) = lim r → r n ω n Z B r ( x ) f ( z ) dz if this limit exists0 otherwise . (2.4)For f i , f ∈ BV ( R n ) , i ∈ N , we say that f i weakly ∗ converges in BV ( R n ) to f if f i converges to f in L ( R n ) and D f i weakly ∗ converges to D f in R n .A measurable subset E of R n is said to be of finite perimeter if its characteristic function χ E is abounded variation function in R n . Let C n denote the set of compact sets of finite perimeter in R n withnonempty interior. The perimeter of E in R n denoted by P ( E ) is defined by P ( E ) = | D χ E | ( R n ) . (2.5)The Hausdorff distance of the sets E , F ∈ C n is defined by d H ( E , F ) = min { λ ≥ K ⊂ L + λ B n , L ⊂ K + λ B n } . For K , L ∈ K n , the Hausdorff distance of K and L (see, e.g., [53, Lemma 1.8.14]) d H ( K , L ) = sup u ∈ S n − | h K ( u ) − h L ( u ) | . For E , F ∈ C n , we define L or also the symmetric-difference distance (sometimes also called Nikod´ymdistance, as in [24]) as follows, d ( E , F ) = L n ( E △ F ) . Let E ∈ C n and let D i χ E denote the i -th component of the distributional gradient D χ E . We denote by ν Ei , i = , . . . , n , the derivative of the measure D i χ E with respect to | D χ E | . The reduced boundary ∂ ∗ E of E is the set of all points x ∈ R n such that the vector ν E ( x ) = ( ν E ( x ) , . . . , ν En ( x )) exists and satisfies | ν E ( x ) | =
1. The vector ν E ( x ) is called the generalized inner normal to E at x . Throughout this paper, let ν Ex ′ and ν Ey denote the front n − n -th component of ν E , respectively.We have (see [1, Theorem 3.59]) D χ E = ν E H n − x ∂ ∗ E . (2.6) he Petty projection inequality for sets of finite perimeter 5 Eq. (2.6) implies that | D χ E | = H n − x ∂ ∗ E (2.7)and that | D i χ E | = | ν Ei |H n − x ∂ ∗ E , i = , . . . , n . For any E ∈ C n and u ∈ S n − , define, for x ′ ∈ u ⊥ , E x ′ , u = { x ′ + tu : t ∈ R , x ′ + tu ∈ E } and ℓ E , u ( x ′ ) = | E x ′ , u | . For u = e n , let E x ′ and ℓ E denote E x ′ , e n and ℓ E , e n , respectively. E will be suppressed when clear from thecontext, and thus we will often denote ℓ E ( x ′ ) by ℓ ( x ′ ) .Let π u ( E ) denote the orthogonal projection of E ⊂ R n onto u ⊥ . In what follows, the essential projectionof a set E ⊂ R n onto u ⊥ is defined as π u ( E ) + = { x ′ ∈ u ⊥ : ℓ E , u ( x ′ ) > } . For u = e n , let π n − ( E ) and π n − ( E ) + denote π e n ( E ) and π e n ( E ) + , respectively.The following theorem is a special case of the co-area formula for rectifiable sets (see [1, Theorem2.93 and Remark 2.94]). Theorem 2.1
Let E ∈ C n . Let g : R n − → [ , + ∞ ) be any Borel function. Then Z ∂ ∗ E g ( x ) | ν Ey ( x ) | d H n − ( x ) = Z R n − dx ′ Z ( ∂ ∗ E ) x ′ g ( x ′ , y ) d H ( y ) . (2.8)Next, we give a theorem concerning one-dimensional sections of sets of finite perimeter, it can beeasily deduced from [1, Theorem 3.108]. Theorem 2.2
Let E be a set of finite perimeter in R n . Then, for L n − -a.e. x ′ ∈ R n − ,E x ′ has f inite perimeter in R y , (2.9) ( ∂ ∗ E ) x ′ = ∂ ∗ ( E x ′ ) , (2.10) ν Ey ( x ′ , y ) = f or every y such that ( x ′ , y ) ∈ ∂ ∗ E , (2.11) lim η → y + χ ∗ E ( x ′ , η ) = , lim η → y − χ ∗ E ( x ′ , η ) = ν Ey ( x ′ , y ) > , lim η → y + χ ∗ E ( x ′ , η ) = , lim η → y − χ ∗ E ( x ′ , η ) = ν Ey ( x ′ , y ) < . (2.12) In particular, a Borel set G E ⊂ π n − ( E ) + exists such that L n − ( π n − ( E ) + \ G E ) = and (2.9)-(2.12) arefulfilled for every x ′ ∈ G E . Y. Lin
Remark 2.1
By [1, Prop. 3.52], if E ⊂ R is a measurable set, then E has finite perimeter in R if and onlyif there exist − ∞ ≤ a < b < · · · < b m ≤ + ∞ such that E = m [ i = ( a i , b i ) up to a set of measure zero.In Theorem 2.2, if E is a set of finite perimeter in R n and ℓ E ( x ′ ) < ∞ for L n − -a.e. x ′ ∈ R n − , then forevery x ′ ∈ G E , E x ′ has finite perimeter in R nx ′ and E x ′ is bounded. Thus, E x ′ is equivalent to { x ′ } × m ( x ′ ) [ i = ( a i , b i ) where m ( x ′ ) = H ( ∂ ∗ ( E x ′ )) , ( x ′ , a i ) , ( x ′ , b i ) ∈ ∂ ∗ ( E x ′ ) and ν Ey ( x ′ , a i ) > and ν Ey ( x ′ , b i ) < . The approximation theorem was proved by Maggi in [49, Theorem 2.4].
Theorem 2.3 (Density of smooth or polyhedral sets). If E is a set of finite perimeter in R n , then a sequenceE i of bounded open sets with with smooth or polyhedral boundary can be found so that E i → E in the L distance and P ( E i ) → P ( E ) . E ∈ C n and u ∈ S n − , we define the Steiner symmetrization , S u E , of E about the hyperplane u ⊥ as S u E = { x ′ + tu ∈ R n : x ′ ∈ π u ( E ) , | t | ≤ ℓ E , u ( x ′ ) / } . For u = e n , let SE (or E s ) denote S e n E .For E ∈ C n , we define the spherical symmetrization , E ⋆ , of E is the close ball centered at the originwhich has the same Lebesgue measure as E .Specially, for K ∈ K n , we have K s = (cid:26)(cid:18) x ′ , y + y (cid:19) ∈ R n − × R y : ( x ′ , y ) , ( x ′ , − y ) ∈ K (cid:27) . In this paper, we shall make critical use of the following fact that was provided by Lutwak et al.in [48, Lemma 1.1].
Lemma 2.1
Suppose K , L ∈ K n and consider K , L ⊂ R n − × R y . ThenSK ∗ ⊂ L ∗ , if and only if h K ( x ′ , t ) = = h K ( x ′ , − s ) , with t = − s = ⇒ h L ( x ′ , t / + s / ) ≤ . In addition, if SK ∗ = L ∗ , then h K ( x ′ , t ) = = h K ( x ′ , − s ) , with t = − s implies that h L ( x ′ , t / + s / ) = . he Petty projection inequality for sets of finite perimeter 7 The following properties of Steiner symmetrization were summarized by Talenti in [54, P.107].
Theorem 2.4
If E ∈ C n , then E s ∈ C n and P ( E s ) ≤ P ( E ) . For E ∈ C n and u ∈ S n − . Let ∂ ∗ u E = { x ∈ ∂ ∗ E : u · ν E ( x ) = } and T ( E ) = { u ∈ S n − : H n − ( ∂ ∗ u E ) = } . (3.1) Theorem 3.1
Let E ∈ C n . Then there exits { u i } ∞ i = ⊂ S n − such that u ∈ T ( E ) , u i ∈ T ( E i − ) when i ≥ ,where E i = S u i . . . S u E, and there exists a subsequence of { E i } , denoted by { E i j } , satisfying E i j → E ⋆ inthe Hausdorff distance. In order to prove Theorem 3.1, we need the following Lemmas 3.1-3.3.
Lemma 3.1 [34, Lemma 6.12] Let E ∈ C n and T ( E ) as in (3.1). Then H n − (cid:0) S n − \ T ( E ) (cid:1) = . Lemma 3.2
Let E i ∈ C n , i ∈ N , and let E be a compact set in R n . Ifd H ( E i , E ) → when i → ∞ , then max x ∈ E i | x | → max x ∈ E | x | . Proof.
Since d H ( E i , E ) → i → ∞ , for any ε >
0, there exists a positive integer N ε such that for any i > N ε , E i ⊂ E + ε B n and E ⊂ E i + ε B n . Thus max x ∈ E i | x | ≤ max x ∈ E + ε B n | x | ≤ max x ∈ E | x | + ε and max x ∈ E | x | ≤ max x ∈ E i + ε B n | x | ≤ max x ∈ E i | x | + ε . Therefore, (cid:12)(cid:12)(cid:12)(cid:12) max x ∈ E | x | − max x ∈ E i | x | (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε . This completes the proof of the lemma. ⊓⊔ Y. Lin
Corollary 3.1
Let E, E i , i ∈ N , be compact sets satisfying E i + ⊂ E i andE = ∞ \ i = E i . (3.2) Then max x ∈ E i | x | → max x ∈ E | x | . Proof.
By (3.2) and the compactness of E i , the sequence E i converges to E in the Hausdorff distance. Thisand Lemma 3.2 yield the desired result. ⊓⊔ Lemma 3.3
Let E, E i , i ∈ N , be compact sets satisfying E i + ⊂ E i andE = ∞ \ i = E i . Then SE = ∞ \ i = SE i . (3.3) Proof.
First, we prove that the orthogonal projections of SE and ∩ ∞ i = SE i onto R n − are same, i.e., π n − ( SE ) = π n − ( ∩ ∞ i = SE i ) . (3.4)If x ′ ∈ π n − ( ∩ ∞ i = SE i ) , then for any i , x ′ ∈ π n − ( SE i ) . Since π n − ( E i ) = π n − ( SE i ) , x ′ ∈ π n − ( E i ) . Thusthere exists x i ∈ E i , i = , , . . . , such that the orthogonal projection of x i onto R n − is x ′ . Since E i + ⊂ E i , x i ⊂ E for any i ∈ N . Thus for the sequence { x i } , there exist a subsequence { x i j } and x ∈ R n such that x i j → x when j → ∞ . It is clear that the orthogonal projection of x onto R n − is also x ′ . Since E i + ⊂ E i and the compactnessof E i , we have x ∈ E i for any i . Thus, x ∈ ∩ ∞ i = E i . This implies x ′ ∈ π n − ( ∩ ∞ i = E i ) = π n − ( E ) = π n − ( SE ) . Therefore, π n − ( ∩ ∞ i = SE i ) ⊂ π n − ( SE ) . Moreover, π n − ( SE ) ⊂ π n − ( ∩ ∞ i = SE i ) is clear. Thus, (3.4) is established.Next, we prove that for any x ′ ∈ π n − ( SE ) , ( SE ) x ′ = ( ∩ ∞ i = SE i ) x ′ . (3.5)Since E i + ⊂ E i , ( E i + ) x ′ ⊂ ( E i ) x ′ . By the limit theorem with respect to sequences of measurable sets(see [18, Theorem 1.2 (iv)]), we have | ( SE ) x ′ | = | ( E ) x ′ | = | ( ∩ ∞ i = E i ) x ′ | = | ∩ ∞ i = ( E i ) x ′ | = lim i → ∞ | ( E i ) x ′ | = lim i → ∞ | ( SE i ) x ′ | = | ( ∩ ∞ i = SE i ) x ′ | . This and the symmetry and compactness of SE and ∩ ∞ i = SE i yield (3.5). The desired equality (3.3) nowfollows from (3.4) and (3.5). ⊓⊔ he Petty projection inequality for sets of finite perimeter 9 Proof of Theorem 3.1.
Define Γ E = { S u k . . . S u E : k ∈ N , u ∈ T ( E ) , u ∈ T ( E ) , . . . , u k ∈ T ( E k − ) } , where T ( E ) is defined as in (3.1) and E j = S u j . . . S u E , j = , . . . , k − E ∈ C n , let r E = max x ∈ E | x | , which is the minimal radius of balls centered at the origin that contain E . Let r be the infimum of all r C , where C ∈ Γ E . Then there is a sequence of { C i } ⊂ Γ E so that r C i → r .Obviously, the sequence { C i } is bounded, because each C i ⊂ r E B n . By [53, Theorem 1.8.5], there is asubsequence C i k that converges to a compact set ¯ E in the Hausdorff distance. By Lemma 3.2, r ¯ E = r .Denote r B n by B , it is clear that ¯ E ⊂ B .Next we prove ¯ E = B . Assume it is not true. There is a small open cap U on ∂ B so that U ∩ ¯ E = /0. Forany line ξ such that ξ ∩ U = /0, either ξ ∩ ¯ E = /0 or the line ξ intersects a longer chord in B than in ¯ E ; that is, | ξ ∩ B | > | ξ ∩ ¯ E | . After taking a Steiner symmetrization S u ¯ E for some u ∈ S n − , the symmetrization S u ¯ E fails to intersect both U and a new cap U ′ given by the reflection of U with respect to the hyperplane u ⊥ .One can continue to take symmetrizations with respect to an appropriate finite family of hyperplanes withnormals v , . . . , v m ∈ S n − that generate finitely many caps covering the whole sphere ∂ B and generate acompact set of finite perimeter about origin ˜ E = S v m . . . S v ¯ E so that | ξ ∩ B | > | ξ ∩ ˜ E | for any line suchthat ξ ∩ ∂ B = /0. Thus, r ˜ E < r .By the above analysis, S n − ⊂ S mi = U v i , where U v i is the reflection of U with respect to the hyperplane v ⊥ i . Since the cap U is open and S n − is compact, there exist sufficiently mall δ , . . . , δ m > u i ∈ B δ i ( v i ) ∩ S n − , i = , . . . , m , we have S n − ⊂ S mi = U u i . By the analysis of the above paragraph, for any u i ∈ B δ i ( v i ) ∩ S n − , i = , . . . , m , let ¯ E = S u m . . . S u ¯ E . Then r ¯ E < r . (3.6)For the above δ , δ , . . . , δ m , by Lemma 3.1, there exits u ∈ B δ ( v ) ∩ S n − such that u ∈ ∞ \ k = T ( C i k ) . Similarly, there exists u ∈ B δ ( v ) ∩ S n − such that u ∈ ∞ \ k = T ( S u C i k ) . Continue to take the process, we can get u , . . . , u m such that u j ∈ B δ j ( v j ) ∩ S n − , j = , . . . , m , and u j ∈ ∞ \ k = T ( S u j − . . . S u C i k ) . Denote ˜ C i k = S u m · · · S u C i k . Since C i k → ¯ E in the Hausdorff distance. For any positive integer i , thereexists a positive integer N such that any k > NC i k ⊂ ¯ E + i B n . Thus S u m S u m − . . . S u C i k ⊂ S u m S u m − . . . S u (cid:18) ¯ E + i B n (cid:19) , which implies thatmax (cid:26) | x | : x ∈ S u m S u m − . . . S u (cid:18) ¯ E + i B n (cid:19)(cid:27) ≥ max {| x | : x ∈ S u m S u m − . . . S u C i k } ≥ r . (3.7)By Lemma 3.3, Corollary 3.1 and (3.7), we havemax {| x | : x ∈ S u m S u m − . . . S u ¯ E } = max ( | x | : x ∈ S u m S u m − . . . S u ∞ \ i = (cid:18) ¯ E + i B n (cid:19)) = max ( | x | : x ∈ ∞ \ i = S u m S u m − . . . S u (cid:18) ¯ E + i B n (cid:19)) = lim i → ∞ max (cid:26) | x | : x ∈ S u m S u m − . . . S u (cid:18) ¯ E + i B n (cid:19)(cid:27) ≥ r . Let ¯ E = S u m . . . S u ¯ E . Then r ¯ E ≥ r . This contradicts (3.6).We have shown that for any E ∈ C n , there are S u i · · · S u E ∈ Γ E so that the Hausdorff distance between S u i · · · S u E and the centered ball B can be arbitrarily small.For a sequence of positive numbers ε k → + , there is D : = S u i · · · S u E ∈ Γ E so that d H ( D , B ) < ε .Similarly, there are D : = S u i . . . S u i + D ∈ Γ D so that d H ( D , B ) < ε . In general, for k = , , . . . ,there are D k : = S u ik . . . S u ik − + D k − ∈ Γ D k − so that d H ( D k , B ) < ε k . Continue the process, we can get asequence { D k } ∞ k = and D k → B in the Hausdorff distance. Let E j = S u j S u j − . . . S u E , j ∈ N . Then { D k } is a subsequence of { E j } while { D k } and { E j } satisfy the conclusions of the theorem. ⊓⊔ For E ∈ C n , we define the projection body Π E of E to be the convex set with support function h Π E ( z ) = Z ∂ ∗ E | z · ν E ( x ) | d H n − ( x ) , z ∈ R n . (4.1)It is clear that the function h Π K ( · ) is the support function of a convex body, Π K , that contains the originin its interior. The polar body of Π E will be denoted by Π ∗ E , rather than ( Π E ) ∗ .The following proposition shows that the Petty projection operator Π : C n → K n is continuous when E i converges to E in the L distance and P ( E i ) → P ( E ) . Proposition 4.1
Let E , E i ∈ C n , i ∈ N . If E i → E in the L distance and P ( E i ) → P ( E ) , then Π E i → Π E in the Hausdorff distance. he Petty projection inequality for sets of finite perimeter 11 Proof.
Since E i converges to E in the L distance when i → ∞ , we have χ E i converges to χ E with respect to L ( R n ) . Since | D χ E i | ( R n ) = P ( E i ) and P ( E i ) → P ( E ) , | D χ E i | ( R n ) is uniformly bounded. Hence, by [1, Proposi-tion 3.13] one deduces that χ E i ⇀ χ E weakly ∗ in R n when i → ∞ . Thus D χ E i ⇀ D χ E weakly ∗ in R n when i → ∞ . (4.2)Since P ( E i ) → P ( E ) , | D χ E i | ( R n ) → | D χ E | ( R n ) . (4.3)By (4.2), (4.3) and Reshetnyak continuity theorem (see [1, Theorem 2.39]), we havelim i → ∞ Z R n (cid:12)(cid:12)(cid:12)(cid:12) u · D χ E i ( x ) | D χ E i ( x ) | (cid:12)(cid:12)(cid:12)(cid:12) d | D χ E i | ( x ) = Z R n (cid:12)(cid:12)(cid:12)(cid:12) u · D χ E ( x ) | D χ E ( x ) | (cid:12)(cid:12)(cid:12)(cid:12) d | D χ E | ( x ) . (4.4)By (2.6), (2.7) and (4.4), for any u ∈ S n − , we havelim i → ∞ Z ∂ ∗ E i (cid:12)(cid:12) u · ν E i ( x ) (cid:12)(cid:12) d H n − ( x ) = Z ∂ ∗ E (cid:12)(cid:12) u · ν E ( x ) (cid:12)(cid:12) d H n − ( x ) . (4.5)By the definition (4.1) of Π E and (4.5), lim i → ∞ h Π E i ( u ) = h Π E ( u ) . Since the support functions h Π E i → h Π E pointwise (on S n − ) they converge unifromly (see, e.g., Schneider [53, p. 54]) completing the proof. ⊓⊔ We now demonstrate the affine nature of the Petty projection operator. Our proof follows along thesame lines as that of Lemma 2.6 proved by Lutwak et al. [48].
Proposition 4.2
If E ∈ C n and A ∈ SL ( n ) , then Π ∗ AE = A Π ∗ E . Proof.
We first suppose that P ∈ C n is a polyhedra satisfying H n − ( { x ∈ ∂ ∗ P : x · ν P ( x ) = } ) = . (4.6)Suppose ( n − ) -dimensional faces of P are F , . . . , F m . Let u , . . . , u m be the outer unit normals to thefaces, and let h , . . . , h m denote the support numbers of the faces of P ; i.e., h i = | w i · u i | , where w i ∈ F i . Let V , . . . , V m denote the volumes of the facial cones, so that, V i = n | h i || F i | . By (4.6), h i = i = , . . . , m .For A ∈ SL ( n ) , let P ⋄ = AP = { Ax : x ∈ P } . Let F ⋄ , . . . , F ⋄ m denote the faces of P ⋄ , let u ⋄ , . . . , u ⋄ m be theouter unit normals of the faces of P ⋄ and let h ⋄ , . . . , h ⋄ m denote the corresponding support numbers of P ⋄ .Since A ∈ SL ( n ) , obviously the volumes V ⋄ , . . . , V ⋄ m of the facial cones of P ⋄ are such that V ⋄ i = V i .The face F i parallel to the subspace u ⊥ i is transformed by A into the face F ⋄ i = AF i parallel to ( A − t u i ) ⊥ and thus u ⋄ i = A − t u i / (cid:12)(cid:12) A − t u i (cid:12)(cid:12) . (4.7) For w ⋄ i ∈ F ⋄ i , there exists w i ∈ F i such that w ⋄ i = Aw i . Thus, from (4.7), we have h ⋄ i = | w ⋄ i · u ⋄ i | = | Aw i · u ⋄ i | = | w i · A t u ⋄ i | = | w i · u i | / (cid:12)(cid:12) A − t u i (cid:12)(cid:12) = h i / (cid:12)(cid:12) A − t u i (cid:12)(cid:12) . (4.8)Since h i = h ⋄ i = i = , . . . , m .Now from definition (4.1), the fact that V ⋄ i = V i together with (4.7) and (4.8), definition (4.1) again,we have, for z ∈ R n , h Π AP ( z ) = h Π P ⋄ ( z ) = m ∑ i = | z · u ⋄ i | | F ⋄ i | = n m ∑ i = (cid:12)(cid:12)(cid:12)(cid:12) z · u ⋄ i h ⋄ i (cid:12)(cid:12)(cid:12)(cid:12) | V ⋄ i | = n m ∑ i = (cid:12)(cid:12)(cid:12)(cid:12) z · A − t u i h i (cid:12)(cid:12)(cid:12)(cid:12) | V i | = n m ∑ i = (cid:12)(cid:12)(cid:12)(cid:12) A − z · u i h i (cid:12)(cid:12)(cid:12)(cid:12) | V i | = h Π P ( A − z ) = h A − t Π P ( z ) , showing that Π AP = A − t Π P . By (2.1), Π ∗ AP = A Π ∗ P . This, together with Proposition 4.1 and Theorem2.3, completes the proof. ⊓⊔ [13, Lemma 3.2] Let E ∈ C n . Then ∂ ℓ ∂ x i ( x ′ ) = Z ( ∂ ∗ E ) x ′ ν Ei ( x ′ , y ) | ν Ey ( x ′ , y ) | d H ( y ) , i = , . . . , n − , (5.1) for L n − -a.e. x ′ ∈ π n − ( E ) + .Remark 5.1 An application of Lemma 5.1 and of (2.10) to E s yields, in particular, ∂ ℓ ∂ x i ( x ′ ) = ν E s i ( x ′ , ℓ ( x ′ )) (cid:12)(cid:12) ν E s y ( x ′ , ℓ ( x ′ )) (cid:12)(cid:12) for L n − -a . e . x ′ ∈ π n − ( E ) + . (5.2) Lemma 5.2 [2, Lemma 4.1] Let E be any set of finite perimeter in R n , n ≥ , and let U be any Borelsubset of R n − . Then H n − ( { x ∈ ∂ ∗ E : ν Ey ( x ) = } ∩ ( U × R y )) = if and only if P ( E ; V × R y ) = f or each Borel subset V o f U such that L n − ( V ) = . Lemma 5.3
Let E ∈ C n satisfy (1.2). Then ( Π ∗ E ) s ⊂ Π ∗ E s . (5.3) Proof.
We will be appealing to Lemma 2.1 and thus we begin by supposing that h Π E ( x ′ , t ) = h Π E ( x ′ , − s ) = , with t = − s , or equivalently, by (2.3) and (2.2), that ( x ′ , t ) ∈ ∂Π ∗ E and ( x ′ , − s ) ∈ ∂Π ∗ E . he Petty projection inequality for sets of finite perimeter 13 By (4.1) 12 Z ∂ ∗ E | ( x ′ , t ) · ν E ( x ) | d H n − ( x ) = Z ∂ ∗ E | ( x ′ , − s ) · ν E ( x ) | d H n − ( x ) = . (5.5)By Lemma 2.1, the desired inclusion (5.3) will have been established if we can show that h Π E s (cid:18) x ′ , t + s (cid:19) ≤ . (5.6)By [13, Proposition 4.2], if (1.2) is established, then H n − ( { x ∈ ∂ ∗ E s : ν E s y ( x ) = } ) = . (5.7)Let G E and G E s be the sets associated with E and E s , respectively, as in Theorem 2.2. By G E , G E s ⊂ π n − ( E ) + , L n − ( π n − ( E ) + \ G E s ) = L n − ( π n − ( E ) + \ G E ) =
0, (1.2), (5.7), Lemma 5.2, (4.1), (5.4)and (5.5), we have that (5.6) is equivalent to Z ∂ ∗ E s ∩ ( G Es × R ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) x ′ , t + s (cid:19) · ν E s ( x ) (cid:12)(cid:12)(cid:12)(cid:12) d H n − ( x ) (5.8) ≤ Z ∂ ∗ E ∩ ( G E × R ) | ( x ′ , t ) · ν E ( x ) | d H n − ( x ) + Z ∂ ∗ E ∩ ( G E × R ) | ( x ′ , − s ) · ν E ( x ) | d H n − ( x ) . We have the following chain of equalities: Z ∂ ∗ E s ∩ ( G Es × R ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) x ′ , t + s (cid:19) · ν E s ( x ) (cid:12)(cid:12)(cid:12)(cid:12) d H n − ( x ) (5.9) = Z G Es dx ′ Z ( ∂ ∗ E s ) x ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) x ′ , t + s (cid:19) · ν E s ( x ) (cid:12)(cid:12)(cid:12)(cid:12) d H ( y ) (cid:12)(cid:12) ν E s y ( x ′ , y ) (cid:12)(cid:12) = Z G E dx ′ Z ( ∂ ∗ E s ) x ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) x ′ , t + s (cid:19) · ν E s x ′ ( x ) (cid:12)(cid:12) ν E s y ( x ′ , y ) (cid:12)(cid:12) , ν E s y ( x ) (cid:12)(cid:12) ν E s y ( x ′ , y ) (cid:12)(cid:12) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d H ( y ) , where the first is due to the co-area formula (2.8) and (2.11), the second to the fact that L n − ( G E s △ G E ) =
0. By (5.2) and (2.10), for E s Z G E dx ′ Z ( ∂ ∗ E s ) x ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) x ′ , t + s (cid:19) · ν E s x ′ ( x ) (cid:12)(cid:12) ν E s y ( x ′ , y ) (cid:12)(cid:12) , ν E s y ( x ) (cid:12)(cid:12) ν E s y ( x ′ , y ) (cid:12)(cid:12) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d H ( y ) (5.10) = Z G E dx ′ Z ∂ ∗ ( E s ) x ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) x ′ , t + s (cid:19) · ∇ ℓ ( x ′ ) , ν E s y ( x ) (cid:12)(cid:12) ν E s y ( x ′ , y ) (cid:12)(cid:12) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d H ( y )= Z G E dx ′ Z ∂ ∗ ( E s ) x ′ ∩ ( G E × R + ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) x ′ , t + s (cid:19) · (cid:18) ∇ ℓ ( x ′ ) , − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) d H ( y )+ Z G E dx ′ Z ∂ ∗ ( E s ) x ′ ∩ ( G E × R − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) x ′ , t + s (cid:19) · (cid:18) ∇ ℓ ( x ′ ) , (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) d H ( y )= Z G E (cid:12)(cid:12)(cid:0) x ′ , t + s (cid:1) · (cid:0) ∇ ℓ ( x ′ ) , − (cid:1)(cid:12)(cid:12) dx ′ + Z G E (cid:12)(cid:12)(cid:0) x ′ , t + s (cid:1) · (cid:0) ∇ ℓ ( x ′ ) , (cid:1)(cid:12)(cid:12) dx ′ . For x ′ ∈ G E , let ∂ r , ∗ E x ′ : = (cid:8) ( x ′ , y ) ∈ ∂ ∗ E x ′ : ν Ey ( x ′ , y ) < (cid:9) (5.11)and ∂ l , ∗ E x ′ : = (cid:8) ( x ′ , y ) ∈ ∂ ∗ E x ′ : ν Ey ( x ′ , y ) > (cid:9) . (5.12)By Remark 2.1, H ( ∂ l , ∗ ( E x ′ )) = H ( ∂ r , ∗ ( E x ′ )) = m ( x ′ ) = H ( ∂ ∗ ( E x ′ )) . (5.13)For k = , . . . , m ( x ′ ) , let y k , l ( x ′ ) be the k -th number y satisfying ( x ′ , y ) ∈ ∂ l , ∗ E x ′ , (5.14) y k , r ( x ′ ) be the k -th number y satisfying ( x ′ , y ) ∈ ∂ r , ∗ E x ′ . (5.15)By (5.1), the monotonicity of | a + br | + | a − br | with respect to r >
0, where a , b ∈ R , (5.13)-(5.15),the last expression of (5.10) ≤ Z G E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x ′ , t + s ) · Z ∂ ∗ E x ′ ν Ex ′ | ν Ey | d H , − Z ∂ ∗ E x ′ d H !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx ′ (5.16) + Z G E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x ′ , t + s ) · Z ∂ ∗ E x ′ ν Ex ′ | ν Ey | d H , Z ∂ ∗ E x ′ d H !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx ′ = Z G E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x ′ , t + s ) · m ( x ′ ) ∑ k = ν Ex ′ | ν Ey | ( x ′ , y k , l ( x ′ )) + ν Ex ′ | ν Ey | ( x ′ , y k , r ( x ′ )) ! , − m ( x ′ ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx ′ + Z G E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x ′ , t + s ) · m ( x ′ ) ∑ k = ν Ex ′ | ν Ey | ( x ′ , y k , l ( x ′ )) + ν Ex ′ | ν Ey | ( x ′ , y k , r ( x ′ )) ! , m ( x ′ ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx ′ . he Petty projection inequality for sets of finite perimeter 15 By the convexity of the absolute value function | · | , (5.11), (5.12) and the co-area formula (2.8), thelast expression of (5.16) ≤ Z G E m ( x ′ ) ∑ k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x ′ , t ) · ν Ex ′ | ν Ey | ( x ′ , y k , r ( x ′ )) , − !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx ′ (5.17) + Z G E m ( x ′ ) ∑ k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x ′ , − s ) · ν Ex ′ | ν Ey | ( x ′ , y k , l ( x ′ )) , !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx ′ + Z G E m ( x ′ ) ∑ k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x ′ , t ) · ν Ex ′ | ν Ey | ( x ′ , y k , l ( x ′ )) , !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx ′ + Z G E m ( x ′ ) ∑ k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x ′ , − s ) · ν Ex ′ | ν Ey | ( x ′ , y k , r ( x ′ )) , − !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx ′ = Z G E dx ′ Z ∂ ∗ E x ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x ′ , t ) · ν Ex ′ | ν Ey | , ν Ey | ν Ey | !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d H ( y )+ Z G E dx ′ Z ∂ ∗ E x ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x ′ , − s ) · ν Ex ′ | ν Ey | , ν Ey | ν Ey | !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d H ( y )= Z ∂ ∗ E (cid:12)(cid:12) ( x ′ , t ) · ν E (cid:12)(cid:12) d H n − + Z ∂ ∗ E (cid:12)(cid:12) ( x ′ , − s ) · ν E (cid:12)(cid:12) d H n − . From (5.9), (5.10), (5.16) and (5.17) we get (5.8). This completes the proof of the lemma. ⊓⊔ Proof of Theorem 1.1.
By Lemma 5.3 and the volume invariance of Steiner symmetrization, inequality(1.3) is established. ⊓⊔ Next, we prove Theorem 1.2. The following lemma was proved in [5, Lemma 3.1] (also see [4]). Wegive a different proof here.
Lemma 5.4
Let E , E i ∈ C n , i ∈ N . If | E | = | E i | and E i → E in the Hausdorff distance, then E i → E in theL distance. Proof.
Since E i → E in the Hausdorff distance, for any ε >
0, there exists a positive integer N such thatfor i > N , E i ⊂ E + ε B n . Thus E i \ E ⊂ ( E + ε B n ) \ E . (5.18)For a decreasing sequence of positive numbers ε k → + , by the limit theorem with respect to sequencesof measurable sets (see [18, Theorem 1.2 (iv)]), we havelim k → ∞ | ( E + ε k B n ) \ E | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ \ k = ( E + ε k B n ) \ E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = . (5.19) By (5.18) and (5.19), lim i → ∞ | E i \ E | = . (5.20)Since | E | = | E i | , we have | E \ E i | = | E | − | E ∩ E i | = | E i | − | E ∩ E i | = | E i \ E | . (5.21)By (5.20) and (5.21), lim i → ∞ d ( E i , E ) = | E i \ E | = ⊓⊔ The Petty projection operator Π : C n → K n is weakly continuous in some sense when E i converges to E ⋆ in the L distance without the assumption that P ( E i ) → P ( E ⋆ ) . Lemma 5.5
Let E , E i ∈ C n , i ∈ N . If L ( E i , E ⋆ ) → , then there exist a subsequence of { Π E i } ∞ i = , denotedby { Π E i } ∞ i = as well, and a convex body K such that ∈ K, d H ( Π E i , K ) → and Π E ⋆ ⊂ K. Proof.
By the definition (4.1) of Π E , (2.5) and (2.7), for any u ∈ S n − , h Π E i ( u ) = Z ∂ ∗ E i | u · ν E i ( x ) | d H n − ( x ) ≤ H n − ( ∂ ∗ E i ) = P ( E i ) . Since P ( E i ) is decreasing with respect to i (see Theorem 2.4), there exists a constant r > Π E i ⊂ r B n for any i . By Blaschke selection theorem (see [53, Theorem 1.8.7]), there exists a subsequenceof { Π E i } ∞ i = , denoted by { Π E i } ∞ i = as well, that converges to a convex body K in the Hausdorff distance.Since h Π E i ( u ) > i and u ∈ S n − and h Π E i ( u ) → h K ( u ) , h K ≥
0. Thus, 0 ∈ K .Since { E i } ∞ i = converges to E ⋆ in the L distance when i → ∞ , we have χ E i converges to χ E ⋆ withrespect to L ( R n ) . Since | D χ E i | ( R n ) = P ( E i ) and P ( E i ) is decreasing with respect to i (see Theorem 2.4), | D χ E i | ( R n ) is uniformly bounded. Hence, by [1, Proposition 3.13] one deduces that D χ E i ⇀ D χ E ⋆ weakly ∗ in R n when i → ∞ . (5.22)By (5.22) and Reshetnyak lower semicontinuity theorem (see [1, Theorem 2.38]), we have Z R n (cid:12)(cid:12)(cid:12)(cid:12) u · D χ E ⋆ ( x ) | D χ E ⋆ ( x ) | (cid:12)(cid:12)(cid:12)(cid:12) d | D χ E ⋆ | ( x ) ≤ lim i → ∞ Z R n (cid:12)(cid:12)(cid:12)(cid:12) u · D χ E i ( x ) | D χ E i ( x ) | (cid:12)(cid:12)(cid:12)(cid:12) d | D χ E i | ( x ) . (5.23)By (2.6), (2.7) and (5.23), for any u ∈ S n − , we have Z ∂ ∗ E ⋆ (cid:12)(cid:12)(cid:12) u · ν E ⋆ ( x ) (cid:12)(cid:12)(cid:12) d H n − ( x ) ≤ lim i → ∞ Z ∂ ∗ E i (cid:12)(cid:12) u · ν E i ( x ) (cid:12)(cid:12) d H n − ( x ) . (5.24)By the definition (4.1) of Π E and (5.24), h Π E ⋆ ( u ) ≤ lim i → ∞ h Π E i ( u ) = h K ( u ) . Thus, Π E ⋆ ⊂ K . ⊓⊔ Proof of Theorem 1.2.
By the monotonicity of the projection operator (Theorem 1.1), the convergence ofthe subsequence { D k } of { E j } (Theorem 3.1 ), the weakly continuity of the projection operator (Lemma5.5), for E ∈ C n , the volume V ( E ) n − V ( Π ∗ E ) is maximized when E = E ⋆ . Since V ( E ⋆ ) n − V ( Π ∗ E ⋆ ) =( ω n / ω n − ) n , inequality (1.4) is established. ⊓⊔ he Petty projection inequality for sets of finite perimeter 17 For E ∈ C n , if E i : = S u i . . . S u E for a sequence of directions { u i } and there exists a sub-sequence E i j converges to E ⋆ in the Hausdorff distance, then does E i converge to E ⋆ in the Hausdorffdistance?If the answer of Problem 6.1 is positive, then the sequence { E i } in Theorem 3.1 converges to E ⋆ ,which is stronger than the convergence of its subsequence. Problem 6.2
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