Anisotropic perimeter and isoperimetric quotient of inner parallel bodies
aa r X i v : . [ m a t h . M G ] J a n ANISOTROPIC PERIMETERAND ISOPERIMETRIC QUOTIENTOF INNER PARALLEL BODIES
GRAZIANO CRASTA
Abstract.
The aim of this note is twofold: to give a short proof of the resultsin [S. Larson,
A bound for the perimeter of inner parallel bodies , J. Funct. Anal.271 (2016), 610–619] and [G. Domokos and Z. Lángi,
The isoperimetric quotientof a convex body decreases monotonically under the eikonal abrasion model ,Mathematika 65 (2019), 119–129]; and to generalize them to the anisotropiccase. Introduction
Let Ω , K ⊂ R n be two convex bodies (i.e., compact convex sets) with non-emptyinterior, and let Ω ∼ λK := { x ∈ R n : x + λK ⊂ Ω } λ ≥ , be the family of inner parallel sets of Ω relative to K , where A ∼ C := T x ∈ C ( A − x ) denotes the Minkowski difference of two convex bodies A and C (see [5, §3.1]). Let r Ω ,K := max { λ ≥ λK + x ⊂ Ω for some x ∈ R n } be the inradius of Ω relative to K , that is, the greatest number λ for which Ω ∼ λK is not empty.For every convex body C ⊂ R n , let P K ( C ) denote its anisotropic perimeter relative to K , defined by(1) P K ( C ) := Z ∂C h K ( ν C ) d H n − , where h K ( ξ ) := sup {h x, ξ i : x ∈ K } is the support function of K , ν C denotes theexterior unit normal vector to C , and H n − is the ( n − -dimensional Hausdorffmeasure. If C is a convex body with non-empty interior, P K ( C ) coincides withthe anisotropic Minkowski content(2) ddt V n ( C + t K ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = lim t → V n ( C + t K ) − V n ( C ) t , Date : January 8, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Inner parallel sets, anisotropic perimeter, isoperimetric quotient. where V n denotes the n -dimensional volume (see [5, Lemma 7.5.3]). Furthermore,in the Euclidean setting (i.e., when K is the unit ball B of R n ), then P B ( C ) = H n − ( ∂C ) .The main results of the present note are Theorems 1.1 and 1.2 below, that havebeen proved in the Euclidean setting in [4, Thm. 1.2] and [3, Thm. 1.1] respectively.We refer the reader to these papers for motivations and applications. Theorem 1.1. (i) Let Ω , K ⊂ R n be two convex bodies with non-empty interior.Then it holds that (3) P K (Ω ∼ λK ) ≥ (cid:18) − λr Ω ,K (cid:19) n − P K (Ω) , ∀ λ ≥ . (ii) Equality holds in (3) for some λ ∗ ∈ (0 , r Ω ,K ) if and only if Ω is homothetic toa tangential body of K . If this is the case equality holds for all λ ≥ and everyparallel set Ω ∼ λK is homothetic to Ω for every λ ∈ [0 , r Ω ,K ) . (We postpone to Section 2 the definition of tangential body.) Theorem 1.2.
Let Ω , K ⊂ R n be two convex bodies with non-empty interior, andlet I ( λ ) := V n (Ω ∼ λK ) P K (Ω ∼ λK ) nn − , λ ∈ [0 , r Ω ,K ) denote the anisotropic isoperimetric quotient of Ω ∼ λK relative to K .Then, either I is strictly decreasing on [0 , r Ω ,K ) , or there is some value λ ∗ ∈ [0 , r Ω ,K ) such that I is strictly decreasing on [0 , λ ∗ ] and constant on [ λ ∗ , r Ω ,K ) .Furthermore, in the latter case, for any λ ∈ [ λ ∗ , r Ω ,K ) , Ω ∼ λK is homothetic bothto Ω ∼ λ ∗ K and to a tangential body of K (more precisely, to an ( n − -tangentialbody of K ). Both results can be interpreted in terms of the level sets of the anisotropicdistance function from the boundary of Ω , defined by(4) δ Ω ,K ( x ) := inf { ρ K ( y − x ) : y ∈ Ω c } , x ∈ Ω where ρ K ( x ) := max { λ ≥ λx ∈ K } is the gauge function of K and we assumethat K contains as an interior point (see [2] for a detailed analysis of δ Ω ,K ).Specifically, since ρ K ( x ) ≤ if and only if x ∈ K , it is not difficult to check that Ω ∼ λK = { x ∈ Ω : δ Ω ,K ( x ) ≥ λ } .We remark that related results in the Euclidean setting are contained in [1, §3],where, in particular, one can find the proof of [4, Thm. 1.2] (see p. 104 andLemma 3.7 therein). NNER PARALLEL BODIES 3 Proof of Theorem 1.1
In the following we shall use the notations of [5]. Let C ⊂ R n be a convex body.We say that x ∈ ∂C is a regular point of ∂C if C admits a unique support planeat x . Given two convex bodies C, K ⊂ R n , we say that C is a tangential body of K if, for each regular point x of ∂C , the support plane of C at x is also a supportplane of K (see [5, §2.2]). From [5, Thm. 2.2.10] it follows that C is a tangentialbody of a ball if and only if it is homothetic to its form body , defined by C ∗ := [ ν ∈ S { x ∈ R n : h x, ν i ≤ } , where S is the set of outward unit normal vectors to ∂C at regular points of ∂C .The definition of p -tangential body is more involved. Since it is not of primaryimportance for the exposition of the paper, we refer to [5, §2.2]. In connectionwith the statement of Theorem 1.2 we limit ourselves to recall that, if C is a p -tangential body of K for some p ∈ { , . . . , n − } , then it is also a tangential bodyof K .Given the convex bodies K , . . . , K n ⊂ R n , we denote by V ( K , . . . , K n ) theirmixed volume (see [5, §5.1]). Moreover, for every pair C, K of convex bodies wedefine V ( i ) ( C, K ) := V ( C, . . . , C | {z } n − i times , K, . . . , K | {z } i times ) , i ∈ { , . . . , n } . From now on we shall assume that Ω , K ⊂ R n are two convex bodies with non-empty interior. To simplify the notation, we denote by r := r Ω ,K the inradius of Ω relative to K , and we define the functions v i ( λ ) := V ( i ) (Ω ∼ λK, K ) , λ ∈ [0 , r ] , i ∈ { , . . . , n } . We recall that, by [5, Lemma 7.5.3], v is differentiable and(5) v ′ ( λ ) = − n v ( λ ) , ∀ λ ∈ [0 , r ] . Theorem 2.1. (i) The functions (6) f i ( λ ) := v i ( λ ) n − i , i ∈ { , . . . , n − } , are concave in [0 , r ] .(ii) Assume that there exists λ ∗ ∈ [0 , r ) such that, for i = 0 or i = 1 , (7) f i ( λ ) = r − λr − λ ∗ f i ( λ ∗ ) , ∀ λ ∈ [ λ ∗ , r ] . Then, for every λ ∈ [ λ ∗ , r ) , Ω ∼ λK is homothetic both to Ω ∼ λ ∗ K , and to atangential body of K .Proof. (i) The claim is a direct consequence of the concavity property of the family λ Ω ∼ λ K (see [5, Lemma 3.1.13]) and of the Generalized Brunn–Minkowskiinequality (see [5, Theorem 7.4.5]). G. CRASTA (ii) Since, by (5), v ′ = − n v = − n f n − and v ( r ) = 0 , if (7) holds for i = 1 then it holds also for i = 0 . Hence, it is enough to prove the claim only in the case i = 0 .Therefore, assume that (7) holds for i = 0 and let λ ∈ [ λ ∗ , r ) . After a translation,we can assume that rK ⊆ Ω , so that ( r − λ ∗ ) K ⊆ Ω ∼ λ ∗ K =: Ω ∗ . Hence r − λr − λ ∗ Ω ∗ = (cid:20) r − λr − λ ∗ Ω ∗ + ( λ − λ ∗ ) K (cid:21) ∼ ( λ − λ ∗ ) K ⊆ Ω ∗ ∼ ( λ − λ ∗ ) K = Ω ∼ λK . On the other hand, (7) implies that the sets r − λr − λ ∗ Ω ∗ and Ω ∼ λK have the samevolume, so that they must coincide, and the conclusion follows. (cid:3) The proof of Theorem 1.1(i) is a direct consequence of Theorem 2.1(i), once werecall that P K ( C ) = n V (1) ( C, K ) (see [5, (5.34)]). Specifically, P K (Ω ∼ λK ) n − = n n − f ( λ ) is a concave (non-negative) function in [0 , r ] , so that (3) follows.Let us prove part (ii). Assume that equality holds in (3) for some λ ∈ (0 , r ) .By the concavity of f it follows that the equality holds in (3) for every λ ∈ [0 , r ] .Hence, the conclusion follows from Theorem 2.1(ii).3. Proof of Theorem 1.2
Using the notation of Section 2, we recall that v ( λ ) := V n (Ω ∼ λK ) = v ( λ ) , p ( λ ) := P K (Ω ∼ λK ) = n v ( λ ) , λ ∈ [0 , r ] . By (5), v is differentiable everywhere with v ′ ( λ ) = − p ( λ ) , whereas p is differen-tiable almost everywhere and admits left and right derivatives at every point, since p n − coincides, up to a constant factor, with the concave function f .Hence, I is right-differentiable at every point of [0 , r ) , and a direct computationshows that its right derivative is given by I ′ + ( λ ) = − p ( λ ) − n +1 n − ξ ( λ ) , λ ∈ [0 , r ) , where(8) ξ ( λ ) := p ( λ ) + nn − v ( λ ) p ′ + ( λ ) . The proof of Theorem 1.2 is then an easy consequence of the following result.
Lemma 3.1.
The function ξ , defined in (8) , is non-negative and non-increasingin [0 , r ) . Furthermore, if ξ vanishes at some point λ ∗ ∈ [0 , r ) , then (7) holds for i = 0 and i = 1 , and, in addition, Ω ∼ λ ∗ K is homothetic to an ( n − -tangentialbody of K . NNER PARALLEL BODIES 5
Proof.
The function ξ ( − λ ) /n coincides with the function ∆( λ ) defined in theproof of Theorem 7.6.19 in [5], where all the stated properties are proved. (cid:3) Remark . In the planar case n = 2 , Theorem 1.2 gives the stronger conclusionthat the isoperimetric quotient is strictly decreasing in [0 , r ) unless Ω is homotheticto K , in which case it is constant. Specifically, assume that ξ ( λ ∗ ) = 0 for some λ ∗ ∈ [0 , r ) ; the stated property will follow if we can prove that Ω = rK . Sincethe only -tangential body to K is K itself, from Lemma 3.1 we deduce that, forevery λ ∈ [ λ ∗ , r ) , Ω ∼ λK is homothetic to K . After a translation we can assumethat Ω ∼ λ ∗ K = ( r − λ ∗ ) K . The concavity property of the family of parallel sets(see [5, Lemma 3.1.13]), together with the fact that Ω ∼ λK = ( r − λ ) K for every λ ∈ [ λ ∗ , r ] , imply that (1 − t )Ω ⊆ (1 − t ) rK ∀ t ∈ [ λ ∗ /r, . For t = λ ∗ /r we get the inclusion Ω ⊆ rK ; on the other hand, the oppositeinclusion Ω ⊇ rK follows from the definition of inradius. References [1] G. Crasta,
Estimates for the energy of the solutions to elliptic Dirichlet problems on convexdomains , Proc. Roy. Soc. Edinburgh Sect. A (2004), no. 1, 89–107. MR2039904[2] G. Crasta and A. Malusa,
The distance function from the boundary in a Minkowski space ,Trans. Amer. Math. Soc. (2007), no. 12, 5725–5759. MR2336304[3] G. Domokos and Z. Lángi,
The isoperimetric quotient of a convex body decreases monotoni-cally under the eikonal abrasion model , Mathematika (2019), no. 1, 119–129. MR3867329[4] S. Larson, A bound for the perimeter of inner parallel bodies , J. Funct. Anal. (2016),no. 3, 610–619. MR3506959[5] R. Schneider,
Convex bodies: the Brunn-Minkowski theory , expanded, Encyclopedia of Math-ematics and its Applications, vol. 151, Cambridge University Press, Cambridge, 2014.MR3155183
Dipartimento di Matematica “G. Castelnuovo”, Univ. di Roma I, P.le A. Moro 2– I-00185 Roma (Italy)
Email address ::