On Volume and Surface Area of Parallel Sets
aa r X i v : . [ m a t h . C A ] O c t ON VOLUME AND SURFACE AREA OF PARALLEL SETS
JAN RATAJ AND STEFFEN WINTER
Abstract.
The r -parallel set to a set A in a Euclidean space consists of allpoints with distance at most r from A . We clarify the relation between the vol-ume and the surface area of parallel sets and study the asymptotic behaviourof both quantities as r tends to 0. We show, for instance, that in general, theexistence of a (suitably rescaled) limit of the surface area implies the existenceof the corresponding limit for the volume, known as the Minkowski content.A full characterisation is obtained for the case of self-similar fractal sets. Ap-plications to stationary random sets are discussed as well, in particular, to thetrajectory of the Brownian motion. Introduction
For r >
0, the r -parallel set A r of a subset A of R d is the set of all points withdistance at most r from A . As r tends to 0, the parallel sets A r approximate theclosure of A . The volume V A ( r ) of A r was investigated by Kneser [10] and later byStach´o [19] who also studied the relation to the ( d − ∂A r . Recently, Hug et al. [9] derived a generalized Steiner formula forclosed sets in R d and obtained as a corollary relations for the volume and surfacearea of parallel sets strengthening those of Stach´o.Also in fractal geometry, parallel sets play an important role. Minkowski contentand Minkowski dimension of A describe the asymptotic behaviour of the volume of A r , as r →
0. The Minkowski dimension (which is equivalent to the box dimension)is an important tool in applications. The Minkowski measurability of a set A , i.e.the existence of its Minkowski content as a positive and finite number, is deeplyconnected with the spectral theory on domains ”bounded” by A , cf. [11] and thereferences therein. For self-similar sets in R d , Gatzouras [7] gave a characterizationof Minkowski measurability and derived formulas for the Minkowski content (in caseit exists) and some suitably averaged counterpart. The idea of approximation withparallel sets has also been used to introduce certain other geometric quantities forfractal sets: Winter [20] and Z¨ahle [21] considered (total) curvature measures of theparallel sets (whenever defined in a generalized sense) and introduced appropriatelyrescaled limits (as r →
0) as fractal curvatures.Parallel sets have also been used to approximate the highly irregular trajectoryof the Brownian motion. Formulas for the mean volume of the parallel sets are
Date : May 28, 2018.2000
Mathematics Subject Classification.
Key words and phrases. parallel set, surface area, Minkowski content, Minkowski dimension,self-similar set, random set.The first author was supported by the Czech Ministry of Education, project no. MSM0021620839. The paper was completed while both authors were supported by a cooperationgrant of the Czech and the German science foundation, GACR project no. P201/10/J039 andDFG project no. WE 1613/2-1. known for decades. Recently, also the mean surface area has been investigated([16]), as well as other curvature functionals ([12], [17]).In this note we investigate more deeply the connection between the volume V A ( r ) and the surface area S A ( r ) := H d − ( ∂A r ) of parallel sets to a set A ⊂ R d ( H d − denotes the ( d − V ′ A ( r ) = S A ( r ), up to countably many r ’s. In Section 3 we use this resultto compare the asymptotic behaviour of surface area and volume. We introducehere, in analogy to the Minkowski content, the surface area based content and sur-face area based dimension which under additional assumptions coincide with theMinkowski’s quantities. To illustrate this relation, consider the case when A is a( d − C smooth compact submanifold of R d . Then, both V A ( r ) / r and S A ( r ) / r →
0, namely to the ( d − A which equals H d − ( A ) in this case. We show that someanalogous results hold for general compact sets with zero volume (and arbitrarydimension). This is closely related to a conjecture that has been communicated byMartina Z¨ahle to the authors: If for a self-similar set A of dimension D the totalcurvatures C k ( A r ) of the parallel sets are defined, then the (appropriately rescaledand averaged) limits of these quantities coincide for all integers k > D −
1. InSection 4, we study the class of self-similar sets more closely and show that theaforementioned surface area based content (the limit of the total curvatures of or-der d −
1) coincides with the Minkowski content provided the set is non-arithmeticwhile the corresponding averaged limits coincide in general; this partially confirmsthe above conjecture. It also extends and sheds a new light on some results in[20]. Finally, Section 5 deals with mean values for stationary random closed sets.As particular applications, we strengthen the results on mean surface area of theparallel set to the Brownian motion from [12] and [16], and derive some estimateson the asymptotics of the surface area.2.
Surface area content of the parallel sets
Let A be a bounded subset of R d and r >
0. Denote by d A be the (Euclidean)distance function of the set A , and by A r and A
We thus have the inequalities( V A ) ′ + ( r ) ≤ M d − ( ∂A Example 2.1. Let A be the union of two unit parallel line segments of distance 2 r in R . Then H ( ∂A r ) = 2+4 πr < πr = H ( ∂A Let C be a totally disconnected compact subset with positive one-dimensional measure of the segment [(0 , , (1 , R and let A = C ∪ ( C + 2 re ),where e = (0 , C + re belongs to the boundary ∂A r , but not to ∂ + A r ,hence, H ( ∂ + A r ) < H ( ∂A r ).Hug et al. claim in [9, p. 257] that ∂A 0, we can assume without loss of generality that A iscompact. Given a point z A , denoteΣ A ( z ) = { a ∈ A : | z − a | = d A ( z ) } , the set of all nearest points of A to z . The point z is called regular if z does notbelong to the convex hull of Σ A . A number r > regular value of d A if allpoints of ∂A For any r > , we have M d − ( ∂A r ) = H d − ( ∂A r ) and M d − ( ∂A With the result of Stach´o we get immediately the following strengthening of [9,Corollary 4.7] and [8, Lemma 1]: Corollary 2.5. The function V A is differentiable at r > with V ′ A ( r ) = H d − ( ∂A r ) = H d − ( ∂ + A r ) = H d − ( ∂A For any r > , H d − ( ∂A r ) ≤ lim s → r − V ′ A ( s ) (the limit is understood over those s < r where V ′ A exists).Proof. Using Corollary 2.4 and (2.1), we get H d − ( ∂A r ) = M d − ( ∂A r ) ≤ M d − ( ∂A Given a compact set A ⊆ R d , we shall use the notation S A ( r ) := H d − ( ∂A r ), r ≥ 0, for the ( d − S A ( r ) as r → V A ( r ) throughCorollary 2.5.Recall the s -dimensional lower and upper Minkowski content of a compact set A ⊂ R d , which are defined by M s ( A ) := lim inf r → V A ( r ) κ d − s r d − s and M s ( A ) := lim sup r → V A ( r ) κ d − s r d − s , where κ t := π t/ / Γ(1 + t ). (If t is an integer, then κ t is the volume of a unit t -ball). If M s ( A ) = M s ( A ), then the common value M s ( A ) is the s -dimensionalMinkowski content of A . We denote bydim M A := inf { t ≥ M s ( A ) = 0 } = sup { t ≥ M s ( A ) = ∞} and dim M A = inf { t ≥ M s ( A ) = 0 } = sup { t ≥ M s ( A ) = ∞} the lower and upper Minkowski dimension of A .In analogy with the Minkowski content, we define for 0 ≤ s < d S s ( A ) := lim inf r → S A ( r )( d − s ) κ d − s r d − − s and S s ( A ) := lim sup r → S A ( r )( d − s ) κ d − s r d − − s . If both numbers coincide, we denote the common value by S s ( A ) and call it the surface area based content or, briefly, the S-content of A . For convenience, weset S d ( A ) := 0. (Note that the above definition would not make sense for s = d .However, setting S d ( A ) zero is justified by the fact that lim r → rS A ( r ) = 0. Indeed,by Corollary 2.4 and [10, Satz 4], we have H d − ( ∂A r ) = M d − ( ∂A r ) ≤ dr ( V A ( r ) − V A (0)) for each r > 0. Since ( V A ( r ) − V A (0)) → r → 0, we obtainlim r → r H d − ( ∂A r ) = 0, as claimed.) N VOLUME AND SURFACE AREA OF PARALLEL SETS 5 We call the numbersdim S A := sup { ≤ t ≤ d : S t ( A ) = ∞} = inf { ≤ t ≤ d : S t ( A ) = 0 } and dim S A := sup { ≤ t ≤ d : S t ( A ) = ∞} = inf { ≤ t ≤ d : S t ( A ) = 0 } the lower and upper surface area based dimension or S-dimension of A , respectively.Obviously, dim S A ≤ dim S A , and if equality holds, the common value will be re-garded as the surface area based dimension (or S-dimension ) of A and denoted bydim S A . Remark. The upper S-dimension dim S A is closely related to the curvature scalingexponent s d − ( A ) defined in [20]. In fact, it is the natural extension of this conceptto arbitrary compact sets (just with a different normalization). Since 2 C d − ( A r ) = H d − ( ∂A r ), whenever C d − ( A r ) is defined, one has dim S A = s d − − d . Similarly, S s ( A ), with s = dim S A , generalizes C fd − ( A ), the fractal curvature of order d − κ d − s ( d − s ). Proposition 3.1. Let A ⊆ R d be compact and let h : [0 , ∞ ) → [0 , ∞ ) be a con-tinuous differentiable function with h (0) = 0 . Assume that h ′ is nonzero on someright neighbourhood of . Let S := lim inf r → S A ( r ) h ′ ( r ) and S := lim sup r → S A ( r ) h ′ ( r ) . Then (3.1) S ≤ lim inf r → V A ( r ) − V A (0) h ( r ) ≤ lim sup r → V A ( r ) − V A (0) h ( r ) ≤ S. In particular, if S = S , i.e. if the limit S := lim r → S A ( r ) h ′ ( r ) ∈ [0 , ∞ ] exists then lim r → V A ( r ) − V A (0) h ( r ) exists as well and equals S .Proof. We follow the lines of the classical proof of l’Hospital’s rule, using the abso-lute continuity of V A (see e.g. [19, Lemma 2]).We shall use the following fact: For any r > there exist < t , t < r suchthat (3.2) ( V A ) ′ ( t ) h ′ ( t ) ≤ V A ( r ) − V A (0) h ( r ) ≤ ( V A ) ′ ( t ) h ′ ( t ) . To see this, fix an r > 0. Since the functionΦ( t ) := ( V A ( r ) − V A (0)) h ( t ) − h ( r ) V A ( t ) , ≤ t ≤ r, is absolutely continuous and Φ(0) = Φ( r ), we have R r Φ ′ ( t ) dt = 0. Hence, eitherΦ ′ ( t ) = 0 for almost all t ∈ (0 , r ), or there exist t , t ∈ (0 , r ) such that Φ ′ ( t ) > > Φ ′ ( t ). This proves (3.2). JAN RATAJ AND STEFFEN WINTER Note that V ′ A ( t ) = S A ( t ) whenever V ′ A ( t ) exists. Thus, taking the lim sup as r → r → V A ( r ) − V A (0) h ( r ) ≤ lim sup r → S A ( t ( r )) h ′ ( t ( r )) ≤ lim sup r → S A ( r ) h ′ ( r ) = S, which is the right inequality in (3.1). Analogously, the left inequality is obtainedby taking the lim inf as r → S = S , then the existence of the limit lim r → V A ( r ) − V A (0) h ( r ) follows immediatelyfrom (3.1). (cid:3) Proposition 3.1 yields the following general relations between Minkowski contentand S-content. Corollary 3.2. Let A ⊂ R d be compact and assume that V A (0) = 0 . Then, for all s ≤ d , S s ( A ) ≤ M s ( A ) ≤ M s ( A ) ≤ S s ( A ) . Proof. In the case s = d , we have S d ( A ) = S d ( A ) = 0 by definition, and it followsfrom the assumption and the continuity of the volume function that M d ( A ) = M d ( A ) = 0 as well.Fix now s < d and let h ( t ) = κ d − s t d − s . Applying Proposition 3.1, we get S s ( A ) = lim inf r → S A ( r ) h ′ ( r ) ≤ lim inf r → V A ( r ) h ( r ) = M s ( A ) . The relation M s ( A ) ≤ S s ( A ) is obtained analogously by applying the third in-equality from (3.1). (cid:3) It is obvious, that the middle inequality in Corollary 3.2 can be strict. Thereare many sets for which the Minkowski content does not exist. However, it was notimmediately clear, whether the left and right inequalities can be strict or whether,in fact, equality holds in general. The following example illustrates that all threeinequalities in Corollary 3.2 can be strict. Example 3.3. The Sierpinski gasket F is the self-similar set in R generated bythe three similarities Φ ( x ) = x , Φ ( x ) = x + ( , 0) and Φ ( x ) = x + ( , √ ).It is well known that its Minkowski dimension is D := dim M F = ln 3ln 2 . We computeits upper and lower ( D -dim.) Minkowski contents and S-contents directly. It turnsout that all four values are different, providing an example where all inequalities inCorollary 3.2 are strict.Observe that the diameter of F is one and that the inradius u of the middle cutout triangle is u := √ . It is not difficult to see that for n ∈ N and r ∈ I n :=[2 − n u, n − u ), the area and boundary length of F r are given by V ( r ) = (cid:18) π − √ n − (cid:19) r + 3 (cid:18) (cid:19) n r + √ (cid:18) (cid:19) n − n − and S ( r ) = (cid:16) π − √ n − (cid:17) r + 3 (cid:18) (cid:19) n . N VOLUME AND SURFACE AREA OF PARALLEL SETS 7 Let t n ( α ) := α − n u , α ∈ [1 , I n . We have t n ( α ) − D = α − D u − D (cid:0) (cid:1) n and thus S ( t n ( α )) t n ( α ) − D = α D u D π + 3 √ n − √ ! + α D − u D − = α D c n + α D − b, where b := 3 u D − and c n := u D (cid:0) − n ( π + 3 √ − √ (cid:1) . Clearly, if we choose thesequence ( α n ) in [1 , 2] such that α n is the value where the function α D c n + α D − b attains its maximum in [1 , n →∞ α Dn c n + α D − n b = lim n →∞ S ( t n ( α n )) t n ( α n ) − D = (2 − D ) κ − D S D ( F ) . Moreover, since c n → c := − √ u D as n → ∞ and since the function g : R → R , ( x, y ) x D y + x D − b is continuous, we have lim n →∞ g ( α n , c n ) = g ( α max , c ),where α max = lim n →∞ α n is the (unique) value where the maximum of the function g ( x, c ) in [1 , 2] is attained. A simple calculation shows that α max = 4(1 − D ). Hence, κ − D S D ( F ) = g ( α max , c )2 − D = α D max c + α D − b − D = 3 √ − D (2 − D )( D − (cid:18) − D (cid:19) D ≈ . . Choosing the sequence ( α n ) such that the minima are attained, a similar argumentshows that α min = 1 and hence κ − D S D ( F ) = g ( α min , c )2 − D = c + b − D = √ − D − D ≈ . . For upper and lower Minkowski content we can argue in the same way. We have V ( t n ( α )) t n ( α ) − D = α D c n + α D − b + α D − b, with b and c n as above. Now we consider the function h : R → R , ( x, y ) x D y + x D − b + x D − b . Choosing α n such that the maximum of h ( x, c n ) in [1 , κ − D M D ( F ) = lim n →∞ V ( t n ( α n )) t n ( α n ) − D = lim n →∞ h ( α n , c n ) = h ( α max , c ) , where c = lim n c n = − √ u D and α max = lim n α n is the value, where h ( x, c )attains its maximum in [1 , 2] (similarly for the minima and the lower Minkowskicontent). It turns out that α max / min = D (cid:16) D − ± q D − D + 1 (cid:17) and thuswe obtain κ − D M D ( F ) = h ( α max , c ) = . . . ≈ . κ − D M D ( F ) = h ( α min , c ) = . . . ≈ . . This shows that for the Sierpinski gasket F we have the strict inequalities S D ( F ) < M D ( F ) < M D ( F ) < S D ( F ) . JAN RATAJ AND STEFFEN WINTER The relations between Minkowski content and S-content in Corollary 3.2 areobtained only in the case when V A (0) = 0. Indeed, it is easy to see that e.g. forthe unit square we have Minkowski dimension equal to 2, while the S-dimensionequals 1. This could be repaired by replacing the volume V A ( r ) with the difference V A ( r ) − V A (0) in the definitions of the Minkowski content and dimension; we shall,however, not follow this way here. Note that in both our applications, self-similarfractal sets and Brownian path, the sets considered have zero volume.For the dimensions, it is clear from the definitions that, in general, one hasdim S A ≤ dim S A ≤ d and dim M A ≤ dim M A ≤ d . From Corollary 3.2 we get Corollary 3.4. For A ⊂ R d compact, we have (i) dim S A ≤ dim M A , (ii) dim M A ≤ dim S A , provided V A (0) = 0 .Proof. If dim M A = d , assertion (i) is obvious. So assume dim M A < d (whichimplies V A (0) = 0). For each dim M A < s ≤ d , we have, by Corollary 3.2, S s ( A ) ≤M s ( A ) = 0 and hence dim S A = inf { t : S t ( A ) = 0 } ≤ s . Since this holds for s arbitrary close to dim M A , we get dim S A ≤ dim M A as claimed.If V A (0) = 0, assertion (ii) follows by a similar argument as (i). (cid:3) In the following, we shall show that even dim M A = dim S A holds whenever V A (0) = 0. Lemma 3.5. If ≤ s < d then lim sup r → V A ( r ) r d − s ≥ d − sd lim sup r → S A ( r )( d − s ) r d − s − . Proof. Using Corollary 2.6, we find that there exists a sequence ( r i ) of differentia-bility points of V A decreasing monotonely to 0 and such thatlim i →∞ V ′ A ( r i )( d − s ) r d − s − i = lim sup r → S A ( r )( d − s ) r d − s − =: a ∈ [0 , ∞ ] . For all r i +1 ≤ r ≤ r i such that V ′ A ( r ) exists (which is the case for H -a.a. r ), wehave V ′ A ( r i ) r d − i ≤ V ′ A ( r ) r d − ≤ V ′ A ( r i +1 ) r d − i +1 , see [19, Theorem 1]. Hence, V A ( r i ) = Z r i V ′ A ( r ) dr = ∞ X j = i Z r j r j +1 V ′ A ( r ) dr ≥ ∞ X j = i Z r j r j +1 V ′ A ( r j ) r d − r d − j dr = ∞ X j = i V ′ A ( r j ) r dj − r dj +1 dr d − j = ∞ X j = i V ′ A ( r j )( d − s ) r d − s − j d − sd r dj − r dj +1 r sj ≥ ∞ X j = i V ′ A ( r j )( d − s ) r d − s − j d − sd ( r d − sj − r d − sj +1 ) . N VOLUME AND SURFACE AREA OF PARALLEL SETS 9 If a ′ < a then V ′ A ( r j )( d − s ) r d − s − j ≥ a ′ for all sufficiently large j . Thus, for i large enough,we have V A ( r i ) r d − si ≥ a ′ d − sd P ∞ j = i ( r d − sj − r d − sj +1 ) P ∞ j = i ( r d − sj − r d − sj +1 ) = a ′ d − sd , which completes the proof. (cid:3) Corollary 3.6. Let A ⊂ R d be a compact set. Then, for any ≤ s ≤ d , M s ( A ) ≥ d − sd S s ( A ) . Consequently, dim M A = dim S A whenever V A (0) = 0 . Curiously, the analogous method fails when trying to show that dim M A =dim S A . A weaker reversed inequality can be derived from the isoperimetric in-equality. Proposition 3.7. Let A ⊂ R d be a compact set. Then, for ≤ s ≤ d , S s d − d ( A ) ≥ c ( M s ( A )) d − d , where c is a constant depending only on s and d . Consequently, dim S A ≥ d − d dim M A .Proof. By the isoperimetric inequality (cf. Federer [3, 3.2.43]) and Corollary 2.4,we have for each r > dκ /dd V A ( r ) ( d − /d ≤ M d − ( ∂A r ) = H d − ( ∂A r ) = S A ( r ) . Fix some s ≤ d and set s ′ := d − d s . Dividing by ( κ d − s r d − s ) d − /d = κ ( d − /dd − s r d − − s ′ ,we get for each r > (cid:18) V A ( r ) κ d − s r d − s (cid:19) ( d − /d ≤ dκ /dd κ ( d − /dd − s S A ( r ) r d − − s ′ = c S A ( r )( d − s ′ ) κ d − s ′ r d − − s ′ , with c := ( d − s ′ ) κ d − s ′ dκ /dd κ ( d − /dd − s . We can assume S s ′ ( A ) < ∞ , since the statement is trivial for S s ′ ( A ) = ∞ . Choosea null sequence ( r n ) n ∈ N such that the limes inferior S s ′ ( A ) is attained, i.e. suchthat lim n →∞ S A ( r n )( d − s ′ ) κ d − s ′ r d − − s ′ n = S s ′ ( A ) ∈ [0 , ∞ ) . Then for each a > S s ′ ( A ) and n sufficiently large, we have S A ( r n )( d − s ′ ) κ d − s ′ r d − − s ′ n ≤ a and thus V A ( r n ) κ d − s r d − sn ≤ c a d/ ( d − . Letting n → ∞ , we obtain( M s ( A )) ( d − /d ≤ (cid:18) lim inf n →∞ V A ( r n ) κ d − s r d − sn (cid:19) ( d − /d ≤ c a , and since this holds for all a > S s ′ ( A ), the first inequality follows.The second inequality is an immediate consequence. (cid:3) Corollary 3.2 shows in particular that, for sets of zero volume, the existence ofthe S-content enforces the existence of the Minkowski content and it also determinesits value: If S s ( A ) = S s ( A ) for some s ≤ d , then also M s ( A ) = M s ( A ) and thecommon value is M s ( A ) = S s ( A ). In particular, if 0 < S s ( A ) < ∞ for some s < d ,then the set A ⊂ R d is Minkowski measurable , i.e. 0 < M s ( A ) < ∞ . Note that ourresults do not allow the converse conclusion. The existence of Minkowski contentdoes not seem to imply the existence of the S-content. Remark. In fact, and as pointed out by the referee, Proposition 3.1 and Lemma 3.5(and also Lemmas 4.1 and 4.6 in the next section) remain true in the slightly moregeneral and purely analytic setting of Kneser functions. A real continuous function f on [0 , ∞ ) is a Kneser function if f ( λb ) − f ( λa ) ≤ λ n ( f ( b ) − f ( a )) , < a ≤ b, λ ≥ . The volume function V A of any bounded set A ⊂ R d is a Kneser function, cf. [10, 19].All the four above mentioned results can be formulated for Kneser functions insteadof V A , since all the properties of V A used in the proofs are consequences of the Kneserproperty. As an illustration, we reformulate here Proposition 3.1: Let f be a Kneser function and let h : [0 , ∞ ) → [0 , ∞ ) be a continuous differ-entiable function with h (0) = 0 . Assume that h ′ is nonzero on some right neigh-bourhood of . Let S := lim inf r → f ′ ( r ) /h ′ ( r ) and S := lim sup r → f ′ ( r ) /h ′ ( r ) .Then S ≤ lim inf r → f ( r ) − f (0) h ( r ) ≤ lim sup r → f ( r ) − f (0) h ( r ) ≤ S. In particular, if S = S , i.e. if the limit S := lim r → f ′ ( r ) /h ′ ( r ) ∈ [0 , ∞ ] exists then lim r → ( f ( r ) − f (0)) /h ( r ) exists as well and equals S . Remark. Since upper Minkowski and upper S-dimension always coincide, cf. Corol-lary 3.6, it is a natural question to ask whether the same is true for the lowercounterparts, i.e., whether the result obtained in Proposition 3.7 can be improved.After this paper was submitted, further investigations of the second author revealedthat there exist sets whose lower S-dimension is strictly smaller than their lowerMinkowski dimension. Even more, the estimate regarding the lower dimensionsin Proposition 3.7 turned out to be optimal. These results will be discussed in aforthcoming paper. 4. Application to self-similar sets We use the above results to study the asymptotic behaviour as r → S F ( r ) of the parallel sets of self-similar sets F satisfying the openset condition. In particular, we show the existence of S D ( F ), provided the set F is non-arithmetic, and the existence of the corresponding average limit e S D ( F ) ingeneral. Here D denotes the similarity dimension of F .We start with two auxiliary results which apply to arbitrary compact sets A ⊆ R d . Recalling the close relation between S A ( r ) and ( V A ) ′ ( r ), we have the followingestimate. Lemma 4.1. Let A be a compact subset of R d and r > . Then for all r > r , ( V A ) ′− ( r ) ≤ (cid:18) rr (cid:19) d − ( V A ) ′− ( r ) . N VOLUME AND SURFACE AREA OF PARALLEL SETS 11 Proof. Let r > r . For each 0 < t < r − r , V A ( r ) − V A ( r − t ) = Z rr − t ( V A ) ′ + ( s ) ds ≤ t sup r − t ≤ s ≤ r ( V A ) ′ + ( s ) . Since, by Stach´o [19, Theorem 1], s − d ( V A ) ′ + ( s ) is decreasing, we infer that ( V A ) ′ + ( s ) ≤ ( s/r ) d − ( V A ) ′ + ( r ) for all r − t ≤ s ≤ r . Hence V A ( r ) − V A ( r − t ) t ≤ sup r − t ≤ s ≤ r (cid:18) sr (cid:19) d − ( V A ) ′ + ( r ) = (cid:18) rr (cid:19) d − ( V A ) ′ + ( r )and for t → V A ) ′− ( r ) ≤ (cid:18) rr (cid:19) d − ( V A ) ′ + ( r ) ≤ (cid:18) rr (cid:19) d − ( V A ) ′− ( r )as claimed. (cid:3) Applying Corollary 2.4 we obtain Corollary 4.2. Let A be a compact subset of R d and < a < b . Then there is aconstant c > such that for all r ∈ [ a, b ] S A ( r ) ≤ c. Proof. By Corollary 2.4, S A ( r ) = H d − ( ∂A r ) ≤ H d − ( ∂A 0. Hence, by Lemma 4.1, we get for all r ∈ [ a, b ], S A ( r ) ≤ ( V A ) ′− ( r ) ≤ (cid:16) ra (cid:17) d − ( V A ) ′− ( a ) ≤ (cid:18) ba (cid:19) d − ( V A ) ′− ( a ) =: c. (cid:3) Let F ⊂ R d be a self-similar set generated by a function system { S , . . . , S N } of contracting similarities S i : R d → R d with contraction ratios 0 < r i < i =1 , . . . , N . That is, F is the unique nonempty, compact set invariant under the setmapping S ( A ) = S i S i ( A ), A ∈ R d . The set F (or, more precisely, the system { S , . . . , S N } ) is said to satisfy the open set condition (OSC) if there exists a non-empty, open and bounded subset O ⊂ R d such that S i S i O ⊆ O and S i O ∩ S j O = ∅ for all i = j . F (or { S , . . . , S N } ) is said to satisfy the strong open set condition (SOSC), if there exist a set O as in the OSC which additionally satisfies O ∩ F = ∅ .It is well known that OSC and SOSC are equivalent, cf. [18], i.e. for F satisfyingOSC, the open set O can always be chosen such that O ∩ F = ∅ .Let D be the similarity dimension of F , i.e. the unique solution s = D of theequation P Ni =1 r si = 1. For F satisfying OSC, D coincides with the Minkowskidimension of F , dim M F = D . Finally, recall that a self-similar set F is called arithmetic (or lattice ), if there exists some number h > − ln r i ∈ h Z for i = 1 , . . . , N , i.e. if {− ln r , . . . , − ln r N } generates a discrete subgroup of R .Otherwise F is called non-arithmetic (or non-lattice ).From the results of the previous section we immediatly derive Proposition 4.3. Let F be a self-similar set satisfying OSC with similarity di-mension D < d . Then dim S F = D . Moreover, S D ( F ) < ∞ , i.e. r D − d +1 S F ( r ) isuniformly bounded as r → . Proof. The equation dim S F = D follows from Corollary 3.6 and the well knownfact that D = dim M F . The finiteness of S D ( F ) is a consequence of Corollary 3.6and the finiteness of the upper Minkowski content M D ( F ), which is well known forself-similar sets. (cid:3) We will establish below, that for non-arithmetic sets F even dim S F = D holds.Now we consider the S-content S D ( F ) of F . It turns out, that in general thislimit does not exist. As for the Minkowski content, Cesaro averaging improves theconvergence. For a compact set A ⊂ R d and 0 ≤ s < d , we define the average s -dimensional S-content e S s ( A ) by(4.1) e S s ( A ) = lim t → | log t | Z t S A ( r )( d − s ) κ d − s r d − − s d log r provided this limit exists, and we write e S s ( A ) and e S s ( A ) for the correspondingupper and lower average limits. Theorem 4.4. Let F ⊂ R d be a self-similar set satisfying OSC and let D < d beits similarity dimension. Then e S D ( F ) of F exists and coincides with the finite andstrictly positive value (4.2) 1 η Z r D − d R ( r ) dr, where η = − P Ni =1 r Di ln r i and the function R : (0 , → R is given by (4.3) R ( r ) = H d − ( ∂F r ) − N X i =1 (0 ,r i ] ( r ) H d − ( ∂ ( S i F ) r ) . If F is non-arithmetic, then also S D ( F ) of F exists and equals the integral in (4.2) . The proof of Theorem 4.4 is postponed to the end of this section. We firstdiscuss the relation of S D ( F ) and the Minkowski content. If S D ( F ) exists, i.e. if F is non-arithmetic, then both limits coincide. Theorem 4.5. Let F be a non-arithmetic self-similar set satisfying OSC and let D < d be its similarity dimension. Then S D ( F ) = M D ( F ) . It follows, that S D ( F ) > and dim S F = D .Proof. Theorem 4.4 states that S D ( F ) exists if F is non-arithmetic and D < d .Hence, the equality of the contents follows from Corollary 3.2. The equality of thedimensions is a consequence of the fact that M D ( F ) > (cid:3) In the arithmetic case, an analogous result holds for the average contents. Wederive it from the following lemma. Recall the definitions of e S s ( A ) and e S s ( A ) from(4.1). Analogously, the average s -dimensional Minkowski content is given by(4.4) f M s ( A ) = lim t → | log t | Z t V A ( r ) κ d − s r d − s d log r and the corresponding lim sup and lim inf are denoted by f M s ( A ) and f M s ( A ). Lemma 4.6. Let A ⊂ R d be compact and ≤ s < d . Then (i) f M s ( A ) ≥ e S s ( A ) and f M s ( A ) ≥ e S s ( A ) N VOLUME AND SURFACE AREA OF PARALLEL SETS 13 (ii) If M s ( A ) < ∞ , then f M s ( A ) = e S s ( A ) and f M s ( A ) = e S s ( A ) .Proof. For 0 < t ≤ 1, let v s ( t ) := Z t V A ( r ) κ d − s r d − s drr and w s ( t ) := Z t S A ( r )( d − s ) κ d − s r d − s − drr . We show that(4.5) v s ( t ) = w s ( t ) + 1 d − s V A ( t ) κ d − s t d − s − d − s ) κ d − s V A (1) . By Corollary 2.5, we have v s ( t ) = Z t Z r S A ( ρ ) dρ drκ d − s r d − s +1 . Interchanging the order of integration, we get v s ( t ) = 1 κ d − s (cid:20)Z t S A ( ρ ) Z t drr d − s +1 dρ + Z t S A ( ρ ) Z ρ drr d − s +1 dρ (cid:21) = 1( d − s ) κ d − s (cid:20) V A ( t ) (cid:18) t d − s − (cid:19) + Z t S A ( ρ ) (cid:18) ρ d − s − (cid:19) dρ (cid:21) = 1 d − s V A ( t ) κ d − s t d − s + w s ( t ) + 1( d − s ) κ d − s ( − V A ( t ) − V A (1) + V A ( t )) , where we used again the relation V A ( r ) = R r S A ( ρ ) dρ . This proves (4.5).Observe that the third term on the right in (4.5) is constant. It vanishes, whendividing by | log t | and taking the limit as t → ∞ . The second term is non-negative.Let ( t n ) be a null sequence, such thatlim n →∞ w s ( t n ) | log t n | = e S s ( A ) . Then f M s ( A ) ≥ lim sup n →∞ v s ( t n ) | log t n | ≥ e S s ( A ) . Similarly the inequality f M s ( A ) ≥ e S s ( A ) is obtained by choosing a sequence ( t n )such that f M s ( A ) is attained.If M s ( A ) < ∞ holds, then the second term on the right in (4.5) is boundedby a constant. Hence, it vanishes when dividing by | log t | and taking the limit as t → ∞ . The stated equalities follow at once. (cid:3) Theorem 4.7. Let F be a self-similar set satisfying OSC and let D < d be thesimilarity dimension of F . Then e S D ( F ) = f M D ( F ) .Proof. By Theorem 4.4, the average S-content of F exists, i.e. e S D ( F ) = e S D ( F ).Since M D ( F ) < ∞ (as is well known and easily verified), Lemma 4.6 (ii) impliesthe assertion. (cid:3) The proof of Theorem 4.4 is based on the following estimates. Fix a feasible set O satisfying the SOSC, i.e. with O ∩ F = ∅ . Let C := S Ni =1 S i O . The followinglemma gives an upper bound for the growth of the surface area of F r near theboundary of C as r → Lemma 4.8. There exist constants c, γ > such that for all < r ≤ H d − ( ∂F r ∩ ( R d \ C ) r ) ≤ cr d − − D + γ . Proof. Let Σ ∗ := S ∞ n =0 { , . . . , N } n and, for 0 < t ≤ 1, let(4.6) Σ( t ) = { w = w . . . w n ∈ Σ ∗ : r w < t ≤ r w r − w n } , where r w := r w . . . r w n . Similarly, we will use S w := S w ◦ . . . ◦ S w n . For conve-nience, let Σ( t ) for t > τ and set r τ := 1 and S τ := id. Furthermore, let r min := min ≤ i ≤ N r i .For a closed set B ⊆ R d and r > 0, let(4.7) Σ( B, r ) = { w ∈ Σ( ρ − r ) : ( S w F ) r ∩ B = ∅} , where ρ is a constant we will fix later. First we show that there is a constant c ′ > B ) such that for all r > H d − ( ∂F r ∩ B ) ≤ c B, r ) r d − . For r > 0, the relation F r ∩ B = S w ∈ Σ( B,r ) ( S w F ) r ∩ B implies that H d − ( ∂F r ∩ B ) ≤ H d − ( [ w ∈ Σ( B,r ) ∂ ( S w F ) r ) ≤ X w ∈ Σ( B,r ) H d − ( ∂ ( S w F ) r ) ≤ X w ∈ Σ( B,r ) r d − w H d − ( ∂F r/r w ) . By definition of Σ( ρ − r ), a := ρ < r/r w ≤ ρr − =: b . Hence, by Corollary 4.2, H d − ( ∂F r/r w ) is bounded by some constant c > r > w ∈ Σ( ρ − r ). Since r w ≤ ρ − r , we obtain H d − ( ∂F r ∩ B ) ≤ c X w ∈ Σ( B,r ) ( ρ − r ) d − = c ′ B, r ) r d − , with c ′ := ρ − d c . This completes the proof of (4.8).Now set B := ( R d \ C ) r . To derive an upper bound for the cardinality ofΣ(( R d \ C ) r , r ), we apply [20, Lemma 5.4.1] with the choice r = 1 and ε = δ . Notethat the set O (1) in [20] equals C . The lemma requires to choose ρ as in [20, (5.1.8),also cf. the paragraph preceding it]. We infer, that there are constants ˜ c, γ > R d \ C ) r , r ) ≤ ˜ cr γ − D for all 0 < r ≤ ρ . By adjusting the constant ˜ c , the estimate can be adapted to holdfor all r ∈ (0 , r ≥ ρ , the cardinality of Σ( B, r ) is bounded by B, ρ ).Now the assertion follows by combining (4.8) and (4.9). (cid:3) Applying Lemma 4.8, we derive the following estimate for the function R in(4.3). Lemma 4.9. There exist c, γ > such that for all < r ≤ | R ( r ) | ≤ cr d − − s + γ . N VOLUME AND SURFACE AREA OF PARALLEL SETS 15 Proof. We abbreviate H := H d − . Fix 0 < r < r min . Set U := S i = j ( S i F ) r ∩ ( S j F ) r and B j := ( S j F ) r \ U . Then F r = S j B j ∪ U is a disjoint union and so H ( ∂F r ) = N X j =1 H ( ∂F r ∩ B j ) + H ( ∂F r ∩ U ) . Similarly, H ( ∂ ( S j F ) r ) = H ( ∂ ( S j F ) r ∩ B j ) + H ( ∂ ( S j F ) r ∩ U ) , since ( S j F ) r ⊆ B j ∪ U . Hence R ( r ) can be written as R ( r ) = N X j =1 (cid:0) H ( ∂F r ∩ B j ) − H ( ∂ ( S j F ) r ∩ B j ) (cid:1) + H ( ∂F r ∩ U ) − N X j =1 H ( ∂ ( S j F ) r ∩ U ) . Observe that ∂F r ∩ B j = ∂ ( S j F ) r ∩ B j . Therefore, all terms of the first sum onthe right are zero. Taking absolute values, we infer(4.10) | R ( r ) | ≤ H ( ∂F r ∩ U ) + N X j =1 H ( ∂ ( S j F ) r ∩ U ) . For the first term note that U ⊆ ( R d \ C ) r (Fact I; see proof below). Recall that C = S Ni =1 S i O . For the remaining terms in (4.10) we have H ( ∂ ( S j F ) r ∩ U ) = r kj H ( ∂F r/r j ∩ S − j U ) ≤ r kj H ( ∂F r/r j ∩ ( R d \ C ) r j /r ) , where the inequality is due to the set inclusion F r/r j ∩ S − j U ⊆ ( O \ R d ) r/r j ⊆ ( R d \ C ) r/r j (Fact II; see proof below). We obtain for each 0 < r ≤ r min ,(4.11) | R ( r ) | ≤ H ( ∂F r ∩ ( R d \ C ) r ) + N X j =1 r d − j H ( ∂F r/r j ∩ ( R d \ C ) r/r j ) . By Lemma 4.8, for each of the terms in (4.11) there are constants c, γ > cr d − − D + γ for 0 < r ≤ r min . Hence wecan also find such constants for | R ( r ) | . The estimate can be adapted to hold forall 0 < r ≤ c , since, by Corollary 4.2, each of the terms of R ( r ) in (4.3) is bounded by a constant for all r ∈ [ r min , Proof of Fact I ( U ⊆ ( R d \ C ) r ) : Let x ∈ U . We show that d ( x, R d \ C ) ≤ r and thus x ∈ ( R d \ C ) r . Assume d ( x, R d \ C ) > r . Since the union C = S i S i O isdisjoint, there is a unique j such that x ∈ S j O . Moreover, d ( x, ∂S j O ) > r . Since x ∈ U , there is at least one index i = j such that x ∈ ( S i F ) r and consequently apoint y ∈ S i F with d ( x, y ) ≤ r . But then y ∈ S i F ∩ S j O , a contradiction to OSC.Hence, d ( x, R d \ C ) ≤ r . Proof of Fact II ( F r/r j ∩ S − j U ⊆ ( O \ R d ) r/r j ) : Let x ∈ F r/r j ∩ S − j U . Then S j x ∈ U and so there exists at least one index i = j with S j x ∈ ( S i F ) r . Hence d ( S j x, ∂S j O ) ≤ r since otherwise there would exist a point y ∈ S i F ∩ S j O , acontradiction to OSC. Therefore, d ( x, ∂O ) ≤ r/r j , i.e. x ∈ ( O \ R d ) r/r j . (cid:3) To complete the proof of Theorem 4.4, we apply the following slight improvementof Theorem 4.1.4 in [20], in which we replace the assumption of continuity off adiscrete set by continuity almost everywhere. Theorem 4.10. Let F be a self-similar set with ratios r , . . . , r N and similaritydimension D . For a function f : (0 , ∞ ) → R , suppose that for some k ∈ R thefunction ϕ k defined by ϕ k ( r ) = f ( r ) − N X i =1 r ki (0 ,r i ] ( r ) f ( r/r i ) is continuous at Lebesgue almost every r > and satisfies (4.12) | ϕ k ( r ) | ≤ cr k − D + γ for some constants c, γ > and all r > . Then r D − k f ( r ) is uniformly bounded in (0 , ∞ ) and the following holds: (i) The limit lim δ → | ln δ | R δ r D − k f ( r ) drr exists and equals (4.13) 1 η Z ε D − k − ϕ k ( ε ) dε , where η = − P Ni =1 r Di ln r i . (ii) If F is non-arithmetic, then the limit of r D − k f ( r ) as r → exists andequals the expression in (4.13) .Proof. The arguments in the proof of Theorem 4.1.4 in [20] can easily be adapted toderive the statement from the Renewal Theorem in Feller [4, p.363]. The assump-tions on ϕ k ensure that the function z : R → R defined by z ( t ) = e ( k − D ) t ϕ k ( e − t )for t ≥ z ( t ) = 0 for t < z is bounded and continuous Lebesgue a.e. and boundedfrom above and below by some directly Riemann integrable functions, then z isdirectly Riemann integrable, cf. for instance [1, Prop. 4.1, p.118]. Clearly, e − γt isdirectly Riemann integrable.) (cid:3) Proof of Theorem 4.4. Apply Theorem 4.10 with f ( r ) := H d − ( ∂F r ), k := d − ϕ k ( r ) := R ( r ). Lemma 4.9 ensures that the hypothesis (4.12) is satisfied. Thecontinuity of R a.e. follows from the same property of H d − ( ∂A r ) for sets A ⊆ R d (cf. Corollary 2.5 and [19, Lemma 2, p.367]). (cid:3) Applications to random sets A random compact set in R d is a measurable mapping Z : (Ω , A , Pr) → ( K ′ , B ( K ′ )) , where K ′ is the family of all nonempty compact subsets of R d and B ( K ′ ) is theBorel σ -algebra on K ′ equipped with the Hausdorff distance (cf. [14]). Theorem 5.1. Let Z ⊆ R d be a random compact set. If the function r E V Z ( r ) is differentiable at some point r > , then V ′ Z ( r ) = H d − ( ∂Z r ) almost surely and ( E V Z ) ′ ( r ) = E H d − ( ∂Z r ) .Proof. First we show that(5.1) E ( V Z ) ′− ( r ) < ∞ for all r > . N VOLUME AND SURFACE AREA OF PARALLEL SETS 17 Indeed, if (5.1) would not hold for some r > E ( V Z ) ′− ( s ) = ∞ for all s < r , which would imply (using Tonelli’s theorem) E V Z ( r ) = Z r E ( V Z ) ′− ( s ) ds = ∞ , r < r, contradicting the assumptions.Next, let r > E V Z ) ′ ( r ) exists. We show that(5.2) E ( V Z ) ′− ( r ) = ( E V Z ) ′− ( r ) , E ( V Z ) ′ + ( r ) = ( E V Z ) ′ + ( r ) . Choose any 0 < r < r . Since V Z ( r ) − V Z ( r − t ) = R rr − t ( V Z ) ′− ( s ) ds for any t < r − r ,we have V Z ( r ) − V Z ( r − t ) t ≤ sup r − t 0, where the words “almost all” can be dropped in dimension d ≤ 3. (An exact formula for the mean volume of the Wiener sausage is known, see[16] and the references therein.) As a corollary of Theorem 5.1, we can show thatthe above mentioned results are true for all r > Corollary 5.2. Let S r,t be the parallel r -neighbourhood of the trajectory of a stan-dard Brownian motion in R d on the time interval [0 , t ] , where r, t > are arbitraryfixed. Then E H d − ( ∂S r,t ) = ∂∂r E H d ( S r,t )= dκ d r d − d − t r + 4 dπ Z ∞ ϕ d ( y t r ) y ( J d − ( y ) + Y d − ( y )) dy ! , where ϕ d ( z ) = 1 − e − z − ze − z /d and J ν , Y ν are the Bessel functions of first andsecond type, respectively, and order ν . In the sequel, we shall consider the asymptotic behaviour of the volume andsurface area of a Brownian motion Z = S on the time interval [0 , S t with a general t > of a parallel set is well known, both almost surely and in the mean. We have, bothalmost surely and in the mean (see [13])lim r → | log r |H ( Z r ) = π, d = 2 , (5.3) lim r → H d ( Z r ) κ d − r d − = ( d − π, d ≥ . (5.4)From the known integral representation of the mean volume, it has been derived in[16] that lim r → r | log r | E H ( ∂Z r ) = π, d = 2 , (5.5) lim r → E H d − ( ∂Z r )( d − κ d − r d − = ( d − π, d ≥ . (5.6)Equations (5.3) and (5.4) imply immediately that dim M Z = 2 almost surely(for d ≥ Z cannot be derived soeasily. Using the methods from Section 3, we get the following estimates. Proposition 5.3. If d = 2 we have almost surely lim sup r → r | log r |H ( ∂Z r ) ≤ π, (5.7) lim inf r → p | log r |H ( ∂Z r ) ≥ π. (5.8) Hence, ≤ dim S Z ≤ dim S Z = 2 almost surely.For d ≥ , we have almost surely lim sup r → H d − ( ∂Z r )( d − κ d − r d − ≤ ( d − d π, (5.9) lim inf r → H d − ( ∂Z r )( d − κ d − r d − − /d > . (5.10) Hence, − d ≤ dim S Z ≤ dim S Z = 2 almost surely.Proof. In order to obtain (5.7), we use a similar method as in the proof of Lemma 3.5.Assume, to the contrary, that r i → i r i | log r i |H ( ∂Z r i ) > π + 2 ε for some ε > 0. Then we have for i sufficiently large, H ( Z r i ) ≥ ∞ X j = i H ( ∂Z r j ) r j − r j +1 r j = ∞ X j = i r j | log r j |H ( ∂Z r j ) r j − r j +1 r j | log r j | r j + r j +1 r j ≥ ( π + ε ) ∞ X j = i r j − r j +1 r j | log r j |≥ ( π + ε ) ∞ X j = i (cid:18) | log r j | − | log r j +1 | (cid:19) = ( π + ε ) 1 | log r i | N VOLUME AND SURFACE AREA OF PARALLEL SETS 19 (the last inequality follows from the concavity of the function t 7→ | log t | − on[0 , e − / ]). Hence, H ( Z r i ) | log r i | ≥ π + ε for sufficiently large i , which is a contra-diction.The lower bound (5.8) follows by the isoperimetric inequality. Indeed, we have4 π H ( Z r ) ≤ H ( ∂Z r ) , hence, 2 √ π p | log r |H ( Z r ) ≤ p | log r |H ( ∂Z r ) , and (5.8) follows using (5.3).(5.9) follows from Lemma 3.5 and (5.10) can be obtained by using the isoperi-metric inequality again, as in the proof of Proposition 3.7. (cid:3) References [1] S. Asmussen: Applied probability and queues. Wiley, Chichester 1987[2] H. Federer: Curvature measures. Trans. Amer. Math. Soc. (1959), 418–491[3] H. Federer: Geometric Measure Theory. Springer, Heidelberg 1969[4] W. Feller: An introduction to probability theory and its applications. Vol. II. (2nd edition)Wiley, New York 1971[5] S. Ferry: When ε -boundaries are manifolds. Fund. Math. (1976), 199–210[6] J. H. G. Fu: Tubular neighborhoods in Euclidean spaces. Duke Math. J. (1985), 1025–1046[7] D. 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