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<ul><li><p>J. Math. Anal. Appl. 413 (2014) 291310</p><p>Contents lists available at ScienceDirect</p><p>D</p><p>Ea Vb D</p><p>a</p><p>ArReAvSu</p><p>KeLoLoSeDeNoBi</p><p>1.</p><p>nowas</p><p>m1thchThboanfotofonoK</p><p>byde</p><p>*</p><p>00htJournal of Mathematical Analysis andApplications</p><p>www.elsevier.com/locate/jmaa</p><p>ifferentiation of sets The general case</p><p>.V. Khmaladze a, W. Weil b,</p><p>ictoria University of Wellington, School of Mathematics, Statistics and Computer Science, P.O. Box 600, Wellington, New Zealandepartment of Mathematics, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany</p><p>r t i c l e i n f o a b s t r a c t</p><p>ticle history:ceived 17 July 2013ailable online 28 November 2013bmitted by B. Bongiorno</p><p>ywords:cal Steiner formulacal point processt-valued mappingrivative setrmal cylinderfurcation</p><p>In recent work by Khmaladze and Weil (2008) and by Einmahl and Khmaladze (2011), limittheorems were established for local empirical processes near the boundary of compactconvex sets K in Rd . The limit processes were shown to live on the normal cylinder of K , respectively on a class of set-valued derivatives in . The latter result was based onthe concept of differentiation of sets at the boundary K of K , which was developed inKhmaladze (2007). Here, we extend the theory of set-valued derivatives to boundaries Fof rather general closed sets F Rd , making use of a local Steiner formula for closed sets,established in Hug, Last and Weil (2004).</p><p> 2013 Elsevier Inc. All rights reserved.</p><p>Introduction</p><p>The general aim of this work is to describe innitesimal changes in the shape of a set in Rd through an appropriatetion of a derivative set. Namely, if bounded sets F () Rd shrink, as 0, to a given set F , then we want to sayhat is the derivative of F (), at = 0. We hereby extend the approach, which was developed in [9] under convexitysumptions.This line of research is motivated by a class of problems in spatial statistics. To be more precise, consider a set A Rd</p><p>arking the boundary between two regions in Rd which carry two different probability distributions. Given n random points, . . . , n chosen independently from the compound distribution in Rd , the statistical challenge is to draw information aboute geometry of A from the empirical process given by the i . This change set problem is a natural generalization of theange point problem on the real line (where A consists of one point only), a classical problem in statistics (see, e.g., [5,4]).e change set problem is of a more recent nature (cf. [10,11,8,13]). For the case where A = K is the boundary of a convexdy K (a compact convex set in Rd), the local empirical process in the neighborhood of K was studied in Khmaladzed Weil [12] and a Poisson limit result was established, as the neighborhood shrinks. The approach made use of a Steinerrmula for support measures (curvature measures), which sit on the normal bundle of K , and the limit process was shownlive on the corresponding normal cylinder. More recently, Einmahl and Khmaladze [6] proved a central limit theoremr such local empirical processes. The Gaussian limit process which they established sits on certain derivative sets in thermal cylinder. This approach required the notion of derivative of sets in measure, a concept which was developed in</p><p>hmaladze [9].Indeed, if a particular choice of a region K is considered as a hypothesis, then the challenging problem is to distinguish,statistical methods, between this K and a class of possible small deformations K of K . It is natural to describe suchformations K = K () by a set-valued function, approaching K as 0. As a stable trace of the deviation K ()K of</p><p>Corresponding author.E-mail addresses: Estate.Khmaladze@vuw.ac.nz (E.V. Khmaladze), wolfgang.weil@kit.edu (W. Weil).</p><p>22-247X/$ see front matter 2013 Elsevier Inc. All rights reserved.tp://dx.doi.org/10.1016/j.jmaa.2013.11.061</p></li><li><p>292 E.V. Khmaladze, W. Weil / J. Math. Anal. Appl. 413 (2014) 291310</p><p>Kprsh</p><p>neFrmde</p><p>aoucyFoseaidiin</p><p>EvCostar(scocoSeonThnadi</p><p>2.</p><p>m</p><p>an</p><p>Th</p><p>Itu(</p><p>WN</p><p>Ththwon() from K , it is consequent to establish a derivative of K () at K as a set in a properly chosen domain. The local pointocesses in the neighborhood of the boundary A = K will live asymptotically on the class of such derivative sets, as wasown in [6,12]. Derivative sets of this type are of interest in innitesimal image analysis in general.It should be mentioned that the differentiation of set-valued functions is a well-established eld of research and promi-nt concepts, much older than that of [9], exist. In particular, the tangent cone approach is described in Aubin andankowska [2] and Borwein and Zhu [3] and provides a classical tool in this eld. A much advanced form of aneappings, the multi-ane mappings of Artstein [1] along with the quasiane mappings of Lemarchal and Zowe [14]monstrate another approach to the differentiability of sets.So far, in the papers [9,6,12] mentioned above, the basic set K was assumed to be compact and convex. This provided</p><p>convenient geometric situation. The set had a well dened outer and inner part, each boundary point had at least oneter normal, the boundary and the normal bundle had nite (d 1)-dimensional Hausdorff measure Hd1, the normallinder had an unbounded upper part and a bounded lower part, and the support measures were nite and nonnegative.r applications, of course, more general set classes would be interesting. Some generalizations, for example to polyconvexts (nite unions of convex bodies) or to sets of positive reach, are possible with minor modications. In the following, wem for a rather general framework allowing closed sets with only few topological regularity properties and we discuss thefferentiation of such sets in the spirit of [9]. In the background is a general Steiner formula for closed sets, established[7], which we will use intensively.General closed sets F Rd can have quite a complicated structure. They need not have a dened inner and outer part.en in the compact case, their boundary can have innite Hausdorff measure Hd1( F ) or even positive Lebesgue measured( F ) > 0. Boundary points x F need not have any normal, but also can have one, two or innitely many normals.nsequently, the normal bundle Nor(F ) of F (or Nor( F ) of F ), as it was dened in [7], can also have a rather complicatedructure. Moreover, the support measures of F , which were introduced in [7] as ingredients of the general Steiner formula,e signed Radon-type measures. They are nite only on sets in the normal bundle with local reach bounded from belowee Section 2, for detailed explanations). In our attempt to dene the derivative of a family F () at a set F , we thereforencentrate on two important situations, which simplify the presentation but are still quite general. First, in Section 3, wensider compact sets F which are the closure of their interior and satisfy d( F ) = 0. We call these solid sets. Second, inction 4, we discuss boundary sets F . These are compact sets without interior points and with d(F ) = d( F ) = 0. Basedthese two set classes, we then study, in Section 5, a differentiation were bifurcation in a set-valued function may occur.e next section, Section 6, investigates some important examples of set functions which are differentiable in our sense,mely families F () which arise as local or global (outer) parallel sets. In the nal section, we discuss some variants of thefferentiability concept. We start in Section 2 with collecting the necessary notations and preliminary results.</p><p>Preliminaries</p><p>In the following, F is a nonempty closed set in Rd and F denotes its boundary. For z Rd , let p(z) = pF (z) be theetric projection of z onto F , that is, the point in F nearest to z, if this point is uniquely determined,z p(z)= min</p><p>xF z x,d let d(z) = dF (z) = z p(z) be the distance from z to F . For > 0, the -neighborhood F of F is dened as</p><p>F ={z Rd: d(z) }.</p><p>e skeleton of F is the set</p><p>S F ={z Rd: a point in F nearest to z is not unique}.</p><p>is known that d(S F ) = 0, where d is the Lebesgue measure in Rd (see [7]). If z / S F F , then p(z) F and we letz) = uF (z) be the corresponding direction, namely the vector in the unit sphere Sd1 given by</p><p>u(z) = z p(z)d(z)</p><p>.</p><p>e call u = u(z) an (outer) normal of F in x = p(z). Note that a point x F can have more than one normal (we denote by(x) the set of all normals in x) and that also some points x F may not have any normal. In that case, we put N(x) = .The (generalized) normal bundle Nor(F ) of F is the subset of F Sd1 dened as</p><p>Nor(F ) = {(x,u): x F , u N(x)}.us, Nor(F ) consists of all pairs (x,u) for which there is a point z / S F F with x = p(z) and u = u(z). Such a point isen of the form z = x+ tu with t = d(z) > 0. Since the ball B(x+ tu, t) touches F only in the point x, this implies that thehole segment [x, x+ tu] projects (uniquely) onto x. This fact gives rise to the reach function r = rF of F , which is denedNor(F ),</p><p>r(x,u) = sup{s > 0: p(x+ su) = x}.</p></li><li><p>E.V. Khmaladze, W. Weil / J. Math. Anal. Appl. 413 (2014) 291310 293</p><p>NItr(co</p><p>N</p><p>Avase{hon(iwh-</p><p>Thsa</p><p>fosu</p><p>co(sSt</p><p>w</p><p>sicato</p><p>si</p><p>re</p><p>ofdo</p><p>Th(xfoFote that in [7], a reach function on Nor(F ) was dened in a slightly different way (by (x,u) = inf{s > 0: x+ su S F }).is easy to see that r and J. Kampf (unpublished) gave an example of a set F and a pair (x,u) Nor(F ) such thatx,u) < (x,u). In the following main result from [7], the local Steiner formula, appeared in the statement in [7], but therrect reach function r was used in the proof.Before we can formulate the result, we need to recall from [7] the notion of a reach measure (F , ) of F . For (x,u) </p><p>or(F ), let h(x,u) [0,] be dened by</p><p>h(x,u) = max{x, r(x,u)1}.subset A Nor(F ) is h-bounded if A {h c}, for some 0 c < . A signed h-measure is then a set function withlues in [,], dened on the system of h-bounded Borel sets in Nor(F ) and such that the restriction of to eacht {h c}, 0 c < , is a signed measure of nite variation. For a signed h-measure, the Hahn-decomposition on each set c} leads to a unique representation = + with mutual singular -nite measures +, 0 which are niteeach sublevel set {h c}, 0 c < . +, and the total variation measure || = + + can then be extended</p><p>n a unique way) to all Borel sets in Nor(F ), but this is not possible, in general, for . Instead of a signed h-measure e speak of an r-measure (reach measure) (F , ) in the following and we call Borel sets A Nor(F ) r-bounded if they arebounded, for the specic function h dened above. We also write ||(F , ) for the variation measure.We denote the minimum of a,b R by a b.</p><p>eorem 1. (See [7].) For any nonempty closed set F Rd, there exist uniquely determined r-measures 0(F , ), . . . ,d1(F , ) of Ftisfying</p><p>Nor(F )</p><p>1B(x)(r(x,u) c)di |i |(F ,d(x,u))< , (1)</p><p>r i = 0, . . . ,d 1, all compact sets B Rd and all c > 0, such that, for any measurable bounded function f :Rd R with compactpport, we have</p><p>Rd\F</p><p>f (z)d(dz) =dj=1</p><p>(d 1j 1</p><p>) Nor(F )</p><p>r(x,u)0</p><p>f (x+ tu)t j1 dtd j(F ,d(x,u)</p><p>). (2)</p><p>The measures 0(F , ), . . . ,d1(F , ) will be called the support measures of F . This notation is justied by the case ofnvex bodies (compact convex sets) F , where the result is well-known and involves the classical support measures of Fee [15]). For convex bodies F the reach function r is innite, r(x,u) = . The local Steiner formula includes the classicaleiner formula (for convex bodies F ),</p><p>d((F + rBd) \ F )= 1</p><p>d</p><p>dj=1</p><p>(d</p><p>j</p><p>)t j d j</p><p>(F ,Nor(F )</p><p>), (3)</p><p>here the total measures i(F ,Nor(F )), i = 0, . . . ,d 1, are proportional to the intrinsic volumes of F .Whereas, for a convex body F , the i(F , ), i = 0, . . . ,d 1, are nite (nonnegative) Borel measures on Nor(F ), the</p><p>tuation is more complicated for closed sets F . As we have explained above, the r-measures i(F , ), i = 0, . . . ,d 1,n attain negative values and are only dened on r-bounded sets, in general. Hence the notion of r-measures is similarthe one of signed Radon measures, as they appear in functional analysis. Since the total variation measure |i |(F , ) =+</p><p>i (F , )+i (F , ) exists on all Borel sets in Nor(F ), the integrability relation (1) guarantees that the integrals on the rightde of (2) exist (without any restriction) and are nite. For more details, see [7].We call a boundary point x F regular, if N(x) consists either of one vector u or of two antipodal vectors u,u. Let</p><p>g(F ) be the set of regular points of F .In the following, we are rst interested in closed sets F , which are solid in the sense that F is nonempty and the closureits interior and that d( F ) = 0 holds. For such sets, we will also develop an expansion into the interior. This can bene simply by replacing F by F , the closure of the complement of F . We have</p><p>Nor( F ) = Nor(F ) Nor(F ), Nor(F ) Nor(F )= .is gives rise to the extended normal bundle Nore(F ) of F which is the union Nor(F ) R(Nor(F )), were R is the reection,u) (x,u). We extend the reach function r of F to the outer reach function r+ on Nore(F ) by putting r+(x,u) = r(x,u),r (x,u) Nor(F ), and r+(x,u) = 0, for (x,u) R(Nor(F ))\Nor(F ). Correspondingly, we dene an inner reach function r ofby r(x,u) = r(F , x,u), for (x,u) R(Nor(F )) and r(x,u) = 0, for (x,u) Nor(F ) \ R(Nor(F )). The support measuresi(F , ), i = 0, . . . ,d 1, of F can be extended to Nore(F ) by putting</p></li><li><p>294 E.V. Khmaladze, W. Weil / J. Math. Anal. Appl. 413 (2014) 291310</p><p>on</p><p>w</p><p>(s</p><p>(sNo[0incoboin</p><p>isre</p><p>HNo</p><p>inre</p><p>3.</p><p>foin</p><p>so</p><p>fo</p><p>fo</p><p>Lei(F , ) = (1)d1ii(F , ) R1</p><p>R(Nor(F )). This denition is consistent since, on the intersection</p><p>Nor(F ) R(Nor(F )),e have</p><p>i(F , ) = (1)d1ii(F , ) R1</p><p>ee [7, Proposition 5.1]).Now the following variant of the local Steiner formula (2) holds,</p><p>Rd\ F</p><p>f (z)d(dz) =dj=1</p><p>(d 1j 1</p><p>) Nore(F )</p><p>r+(x,u)r(x,u)</p><p>f (x+ tu)t j1 dtd j(F ,d(x,u)</p><p>)(4)</p><p>ee [7, Theorem 5.2]). Since we have assumed d( F ) = 0, the integration on the left can be performed over the whole Rd .te that d( F ) = 0 need not even hold, if F is the closure of its interior. An example is given by a Cantor-type set in,1]. As in the classical Cantor set, open intervals are deleted in each step, but such that the total length of all deletedtervals is a constant c < 1. Let A be the union of all open intervals which are deleted in even-numbered steps and B therresponding union of the intervals deleted in odd-numbered steps. A and B are disjoint open sets and their (common)undary is C = [0,1] \ (A B) with 1(C) = 1 c > 0. Moreover, the sets A C and B C are both the closure of theirterior.The summand in (4) for j = 1 involves the support measure d1(F , ). As it follows from [7, Proposition 4.1], d1(F , )a nonnegative -nite measure on Nore(K ) which, for a solid set F , is concentrated on the pairs (x,u) with x rege(F ) =g(F ) reg(F ) and is given by the Hausdorff measure,</p><p>d1(F , ) =</p><p>rege(F )</p><p>1{(x, (F , x)</p><p>) }Hd1(dx). (5)ere, (F , x) is the normal vector u N(x), for which (x,u) Nore(F ) (for x rege(F ), this vector u is uniquely determined).te that Hd1( F \ rege(F )) > 0 is possible, even for solid sets F .For (full dimensio...</p></li></ul>