Stability of boundary measures

apport   de recherche ISSN 0249-6399 ISRN INRIA/RR--6219--FR+ENG Thème SYM INSTITUT N A TION AL DE RECHERCHE EN INFORMA TIQUE ET EN A UTOMA TIQUE Stability of boundary measures Frédéric Chazal — David C ohen-Steiner — Quent in Mérigot N° 6219 14 June 2007 Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France) Téléphone : +33 4 92 38 77 77 — Téléco pie : +33 4 92 38 77 65 Stabilit y of b oundary measures F rédéri Chazal , Da vid Cohen-Steiner , Quen tin Mérigot Thème SYM  Systèmes sym b oliques Pro jet Geometria Rapp ort de re her he n ° 6219  14 June 2007  20 pages Abstrat: W e in tro due the b oundary me asur e at sale r of a ompat subset of the n -dimensional Eulidean spae. W e sho w ho w it an b e omputed for p oin t louds and suggest these measures an b e used for fe atur e dete tion . The main on tribution of this w ork is the pro of a quan titativ e stabilit y theorem for b oundary measures using to ols of on v ex analysis and geometri measure theory . As a orollary w e obtain a stabilit y result for F ederer's urvatur e me asur es of a ompat, allo wing to ompute them from p oin t-loud appro ximations of the ompat. Key-w ords: dimension detetion, p oin t louds, urv ature measures, on v ex funtions, nearest neigh b or. Stabilité de mesures de b ord Résumé : Nous in tro duisons la notion de mesur e de b or d d'é helle r d'un sous-ensem ble ompat de l'espae eulidien de dimension n . Nous mon trons ommen t aluler es mesures p our un n uage de p oin ts et suggérons que es mesures p euv en t être utilisées p our de la détetion de fe atur es . La prinipale on tribution de e tra v ail est la démonstration d'un théorème quan titatif de stabilité des mesures de b ord, utilisan t des outils de l'analyse on v exe et de la théorie géométrique de la mesure. En orollaire, w ous obtenons un résultat de stabilité des mesur es de  ourbur e d'un ompat (notion in tro duite par F ederer), p ermettan t de les aluler à partir d'appro ximations du ompat par des n uages de p oin ts. Mots-lés : détetion de dimension, n uages de p oin ts, mesures de ourbure, fontions on v exes, plus pro  he v oisin. Boundary me asur es 3 In tro dution Motiv ations and previous w ork. The main goal of our w ork is to dev elop a framew ork for fe atur es dete tion : nding the b oundaries, sharp edges, orners of a ompat set K ⊆ R n kno wing only a p ossibly noisy p oin t loud sample of it. This problem has b een an area of ativ e resear h in omputer siene for some y ears. Man y of the urren tly used metho ds for feature and dimension detetion (see [DGGZ03℄ and the referenes therein) rely on the omputation of a V oronoï diagram. The ost of this omputation is exp onen tial in the dimension and annot b e pratially realized for an am bien t dimension m u h greater than three. In lo w dimension, sev eral metho ds ha v e b een in v en ted for b oundary detetion (mostly to detet holes), for example [FK06 ℄ (2D, graph-based), [BSK ℄ (3D), and [RBBK06 ℄. Sharp edges detetion has also b een studied in [GWM01 ℄, and reen tly in [DHOS07℄. The algorithms w e dev elop ha v e three main adv an tages: they are built on a strong math- ematial theory , are robust to noise and their ost dep end only on the in trinsi dimension of the sampled ompat set. None of the existing metho ds for feature detetion share these three desirable prop erties at the same time. Boundary measures and their stabilit y . Giv en a sale parameter r , w e asso iate to ea h ompat subset K of R n a probabilit y measure β K,r . This b oundary me asur e of K at s ale r as w e all it, giv es for ev ery Borel set A ⊆ R n the probabilit y that the pro jetion on K of a random p oin t at distane at most r of K lies in A (the pr oje tion on K , denoted b y p K , maps almost an y p oin t in R n to its losest p oin t in K ). In tuitiv ely , the measure β K,r will b e more onen trated on the fe atur es of K : for instane, if K is a on v ex p olyhedron in R 3 , β K,r will  harge the edges more than the faes, and the v erties ev en more (see example I). It should also b e notied that this measure is losely related to F ederer's urvatur e me asur es (in tro dued in [F ed59 ℄). This artile fo uses on the stabilit y prop erties of the b oundary measures, sho wing that they an b e appro ximated from a noisy sample of K . The problem of extrating geometri information from these b oundary measures will b e treated in an up oming w ork. The main stabilit y theorem an b e stated as follo w: Theorem (IV.1) . If one endows the set of  omp at subsets of R n with the Hausdor dis- tan e, and the set of  omp atly supp orte d pr ob ability me asur es on R n with the W asserstein distan e, the map K 7→ β K,r is lo  al ly 1 / 2 -Hölder. In the sequel w e will mak e this statemen t more preise b y giving expliit onstan ts. A v ery similar stabilit y result for a generalization of F ederer's urvatur e me asur es is dedued from this theorem. W e dedue theorem IV.1 from the t w o theorems I I I.5 and I I.3 b elo w, whi h are also in teresting in their o wn. Theorem (I I I.5) . L et E b e an op en subset of R n with ( n − 1) r e tiable b oundary, and f , g b e two  onvex funtions suh that diam( ∇ f ( E ) ∪ ∇ g ( E )) 6 k . Then ther e exists a  onstant C ( n, E , k ) dep ending only on n and E suh that for k f − g k ∞ smal l enough, k∇ f − ∇ g k L 1 ( E ) 6 C ( n, E , k ) k f − g k 1 / 2 ∞ RR n ° 6219 4 Chazal, Cohen-Steiner & Mérigot Theorem (I I.3) . If K is a  omp at set of R n , for every p ositive r , ∂ K r = { x ; d( x, K ) = r } is ( n − 1) r e tiable and H n − 1 ( ∂ K r ) 6 N ( ∂ K, r ) × ω n − 1 (2 r ) Theorem I I I.5 is used to sho w that the map K 7→ p K ∈ L 1 ( E ) (where p K is the pro jetion on K ) is lo ally 1 / 2 -Hölder, whi h is the main ingredien t for the stabilit y result. Theorem I I.3 impro v es up on [OP85 ℄, in whi h Oleksiv and P esin pro v e the niteness of the measure of the lev el sets of the distane funtion to K . It is used here as a to ol to sho w that K r ∆ K ′ r is small when K and K ′ are lose ( A ∆ B b eing the symmetri dierene b et w een A and B , and K r b eing the set of p oin ts at distane at most r from K ). Outline. In the rst setion w e giv e some examples of b oundary measures and sho w ho w they an b e omputed eien tly for p oin t louds. The seond and third setions on tain the pro ofs of theorems I I.3 and I I I.5 resp etiv ely . In the fourth setion w e dedue from these theorems the stabilit y results for b oundary and urv ature measures. I Denition of b oundary measures Some examples of b oundary measures Notations. If K is a ompat subset of R n , the distane to K is dened as d K ( x ) = min y ∈ K k x − y k . The r -tubular neigh b orho o d or r -oset around a subset F ⊆ R n is the set of p oin ts at distane at most r from F , and is denoted b y F r . F or x ∈ R n , the set of p oin ts y ∈ K that realizes this minim um is denoted b y pro j K ( x ) . One an sho w that # pro j K ( x ) = 1 i d K is dieren tiable at x . Sine d K is 1 -Lips hitz, a theorem of Radema her ensures that b oth onditions are true for almost ev ery p oin t x ∈ R n . This allo ws us to dene a funtion p K ∈ L 1 lo c ( R n ) , alled the pro jetion on K , whi h maps (almost) ev ery p oin t x ∈ R n to its only losest p oin t in K . The s -dimensional Hausdor measure is denoted b y H s ; in partiular H n oinides with the usual Leb esgue measure on R n . Denition I.1. The r -sale b oundary measure β K,r of a ompat K of R n asso iates to an y Borel set A ⊆ R n the probabilit y that the pro jetion of a random p oin t at distane less than r of K lies in A . If w e denote b y µ K,r the pushforw ard of the uniform measure on K r b y the pro jetion on K , ie. for all Borel set A ⊆ R n , µ K,r ( A ) = H n ( p − 1 K ( A ) ∩ K r ) , then β K,r = H n ( K r ) − 1 µ K,r . Examples. 1. If C = { x i ; 1 6 i 6 N } is a p oin t loud , that is a nite set of p oin ts of R n , then β C,r is a sum of w eigh ted Dira measures. Indeed, if V o r C ( x i ) denotes the V oronoi ell of x i , that is the set of p oin ts loser to x i than to an y other p oin t of C , w e ha v e µ C,r = n X i =1 H n (V or C ( x i ) ∩ C r ) δ x i INRIA Boundary me asur es 5 2. Let S b e a unit-length segmen t in the plane with endp oin ts a and b . The set S r is the union of a retangle of dimension 1 × 2 r whose p oin ts pro jets on the segmen t and t w o half-disks of radius r whose p oin ts are pro jeted on a and b . It follo ws that µ S,r = 2 r H 1   S + π 2 r 2 δ a + π 2 r 2 δ b 3. Let P b e a on v ex solid p olyhedron of R 3 , { e j } b e its edges and { v k } b e its v erties. W e denote b y a ( e j ) the angle b et w een the normals of the t w o faes on taining e i , and b y K ( v k ) the solid angle formed b y the normal one at v k . Then one an see that µ P,r = H 3   P + r H 2   ∂ P + X j r 2 a ( e j ) × H 1   e j + X k r 3 K ( v k ) δ v k 4. More generally , if K is a ompat with p ositiv e rea h, in the sense that there exists a p ositiv e r su h that the pro jetion on K is unique for an y p oin t in K r , there exist Borel measures (Φ K,i ) 0 6 i 6 n on R n su h that µ K,r = n X i =0 r n − i ω n − i Φ K,i where ω i is the v olume of the unit sphere in R i +1 . These measures Φ K,i are alled the urvatur e me asur es of the ompat set K and ha v e b een in tro dued under this form b y F ederer in [F ed59℄, generalizing existing notions in the ase of on v ex subsets and ompat smo oth submanifolds of R n (Mink o wski's Quermassinte gr al and W eyl's tub e form ula, f. [W ey39 ℄). The seond and third examples sho w exatly the kind of b eha viour w e w an t to exhibit (and so do es gure I.1): the measure β K,r an b e written as a sum of w eigh ted Hausdor measures of v arious dimension, onen trated on the features of K : its b oundary , its edges and its orners. This remark together with the stabilit y theorem for b oundary measures sho ws that they are a suitable to ol to b e used in robust feature extration algorithms. In the next paragraph w e sho w ho w to ompute them eien tly for p oin t louds. The b oundary measure of a p oin t loud A fast metho d for omputing the b oundary measures of p oin t louds is of ruial imp ortane for pratial appliations. Indeed, most real-w orld data, either 3 D (laser sans) or higher dimensional is giv en in the form of an unstrutured p oin t loud. Sine omputing the V oronoï diagram of a p oin t loud has an exp onen tial ost in the ambient dimension, w e will b e using a probabilisti Mon te-Carlo metho d to get an appro ximation of the b oundary measures. In a v ery general w a y , if µ is an absolutely on tin uous measure on R n , one an ompute p # C µ as sho wn b elo w. The three main steps of this algorithm ( I , I I , and I I I ) are desrib ed with more detail in the follo wing paragraphs. RR n ° 6219 6 Chazal, Cohen-Steiner & Mérigot Input: a p oin t loud C = { x i } , a measure µ Output: an appro ximation of p C # µ in the form P k ( i ) δ X i [ I. ℄ Cho ose N big enough to get a go o d appro ximation with high ondene while n 6 N do [ I I. ℄ Cho ose a random p oin t X n with probabilit y distribution µ [ I I I. ℄ Finds its losest p oin t x i in the loud C , add 1 to n ( x i ) end while return [ n ( x i ) / N ] i . Step I. The measure µ N = 1 / N P i 6 N δ X i where ( X i ) is a sequene of indep enden t random v ariables whose la w are µ is alled an empiri al me asur e . The question of whether (and at what sp eed) µ N on v erge to µ as N gro ws to innit y is w ell-kno wn to probabilists and statistiians. The results of this setion are not original and an probably b e impro v ed, they are presen ted here only to giv e pr o of-of- on ept b ounds for N . Theorem I.2 (Ho eding's inequalit y) . If ( Y i ) is a se quen e of indep endent [0 , 1] -value d r andom variables whose  ommon law ν has a me an m ∈ R , and Y N = (1 / N ) P i 6 N Y i then P (   Y N − m   > ε ) 6 2 exp( − 2 N ε 2 ) In partiular, let's onsider a family ( X i ) of indep enden t random v ariables distributed aording to the la w p C # µ . Then, for an y 1 -Lips hitz funtion f : R n → R with k f k ∞ 6 1 , one an apply Ho eding's inequalit y to the family of random v ariables Y i = f ( X i ) : P "      1 N N X i =1 f ( X i ) − Z f d µ      > ε # 6 2 exp( − 2 N ε 2 ) This kind of estimate also follo ws from T alagrand T 1 ( λ ) -inequalities, in whi h ase the fator 2 in the exp onen tial is replaed b y 2 λ . Bolley , Guillin and Villani use this fat to get quan titativ e onen tration inequalities for empirial measures with non-ompat supp ort in [BGV07 ℄. W e no w let BL 1 ( C ) b e set of Lips hitz funtions f on C whose Lips hitz onstan t Lip f is at most 1 and k f k ∞ 6 1 . W e let N (BL 1 ( C ) , k . k ∞ , ε ) b e the minim um n um b er of balls of radius at most r (with resp et to the k . k ∞ norm) needed to o v er BL 1 ( C ) . Prop osition I.3 giv es a b ound for this n um b er. It follo ws from the denition of the b ounded-Lips hitz distane (see I.4) and from the union b ound that P [ d bL (p C # µ N , p C # µ ) > ε ] 6 2 N (BL 1 ( C ) , k . k ∞ , ε/ 4 ) exp( − N ε 2 / 2) Pr oposition I.3 . F or any  omp at metri sp a e K , N (BL 1 ( K ) , k . k ∞ , ε ) 6  4 ε  N ( K,ε/ 4) INRIA Boundary me asur es 7 Pr o of. Let X = { x i } b e an ε/ 4 -dense family of p oin ts of K with # X = N ( K, ε/ 4) . It is easily seen that for ev ery 1 -Lips hitz funtions f , g on K , k f − g k ∞ 6 k ( f − g ) | X k ∞ + ε/ 2 . Then, one onludes using that N (BL 1 ( X ) , k . k ∞ , ε/ 2 ) 6 (4 /ε ) # X . In ne one gets the follo wing estimate on the b ounded-Lips hitz distane b et w een the empirial and the real measure: P [ d bL (p C # µ N , p C # µ ) > ε ] 6 2 exp  ln(16 / ε ) N ( C, ε/ 16 ) − N ε 2 / 2  Sine C is a p oin t loud, the oarsest p ossible b ound on N ( C, ε/ 16 ) , namely # C , sho ws that omputing an ε -appro ximation of the measure p # µ with high ondene ( e g. 99% ) an b e done with N = O (# C ln(1 / ε ) /ε 2 ) . Step I I. T o sim ulate the uniform measure on K r one annot simply sho ot p oin ts in a b ounding b o x of K r , k eeping those that are atually in K r sine this has an exp onen tial ost in the am bien t dimension. Lu kily there is a simple algorithm to generate p oin ts aording to this la w whi h relies on pi king a random p oin t x i in the loud C and then a p oin t X in B ( x i , r )  taking in to aoun t the o v erlap of the balls B ( x, r ) where x ∈ C : Input: a p oin t loud C = { x i } , a salar r Output: a random p oin t in C r whose la w is H n | K r rep eat Pi k a random p oin t x i in the p oin t loud C Pi k a random p oin t X in the ball B ( x i , r ) Coun t the n um b er k of p oin ts x j ∈ C at distane at most r from X Pi k a random in teger d b et w een 1 and k un til d = 1 return X . Step I I I. The trivial algorithm for omputing the pro jetion of a p oin t on a p oin t loud tak es exatly n steps. Sine generally N will an order of magnitude greater than n w e migh t impro v e the o v erall O ( n 2 ) ost b y main taining a data struture whi h allo ws fast nearest-neigh b our queries. This problem is notoriously diult and un til reen tly most of the eien t algorithms in high dimension w ere only able to ompute appr oximate ne ar est neighb ours . This amoun ts to replaing p C b y a map ˜ p ε with the prop ert y that for all x , k ˜ p ε ( x ) − p C ( x ) k 6 (1 + ε )d C ( x ) . Unfortunately , the te hniques w e dev elop in this pap er do not seem to apply diretly to get quan titativ e loseness estimates for the measures ˜ p ε # µ and p K # µ . It should b e noted that for lo w en trop y p oin t louds, nearest neigh b or queries an b e done more eien tly . F or instane, a reen t artile b y Beygelzimer, Kak ade and Langford ( f. [BKL06 ℄) in tro dues a struture alled  over tr e es whi h allo ws an exat nearest neigh b our query with omplexit y O ( c 12 log n ) where c is related to the in trinsi dimension of the p oin t loud, with an initialisation ost of O ( c 6 n log n ) . RR n ° 6219 8 Chazal, Cohen-Steiner & Mérigot Figure I.1: Boundary measure for a sampled me hanial part. W asserstein distane and stabilit y Sine our goal is to giv e a quan titativ e stabilit y result for b oundary measures, w e need to put a metri on the spae of probabilit y measures on R n . The W asserstein distane, related to the Monge-Kan toro vi h optimal transp ortation problem seemed in tuitiv ely (and later happ ened to really b e) appropriate for our purp oses. A go o d referene on this topi is Cédri Villani's b o ok [Vil03 ℄. Denition I.4. The set of measures (resp. probabilit y measures) on R n is denoted b y M ( R n ) (resp. M 1 ( R n ) ). W e endo w M ( R n ) with the b ounded Lips hitz distane, ie. ∀ µ, ν ∈ M ( R n ) , d bL ( µ, ν ) = sup k ϕ k Lip 6 1     Z ϕ d µ − Z ϕ d ν     where the suprem um is tak en o v er all Lips hitz funtions ϕ with k ϕ k Lip = Lip ϕ + k ϕ k ∞ 6 1 ( Lip ϕ b eing the smallest onstan t k su h that ϕ is k -Lips hitz). W e put t w o distanes on M 1 ( R n ) (whi h are in fat iden ti, see b elo w). The F ortet- Mourier distane, whi h is almost the same as the b ounded Lips hitz one: ∀ µ, ν ∈ M 1 ( R n ) , d FM ( µ, ν ) = sup Lip ϕ 6 1     Z ϕ d µ − Z ϕ d ν     And the W asserstein distane: W 1 ( µ, ν ) = inf { E (d( X, Y )) ; law( X ) = µ, law( Y ) = ν } where the inm um is tak en o v er all random v ariables X and Y whose la ws are µ and ν resp etiv ely . INRIA Boundary me asur es 9 Notations. If µ and ν ∈ M ( R n ) are absolutely on tin uous with resp et to H n , ie. d µ = ϕ d H n and d ν = ψ d H n w e denote b y µ ∩ ν the measure dened b y d( µ ∩ ν ) = min ( ϕ, ψ )d H n , and µ ∆ ν = µ + ν − 2 µ ∩ ν . Pr oposition I.5 . If µ ∈ M ( R n ) is absolutely  ontinuous with r esp e t to the L eb esgue me a- sur e, and f , g : R n → R n ar e two funtions in L 1 ( µ ) , then d bL ( f # µ, g # µ ) 6 k f − g k L 1 ( µ ) If µ and ν ar e two absolutely  ontinuous me asur es on R n , d bL ( f # µ, g # ν ) 6 k f − g k L 1 ( µ ∩ ν ) + mass( µ ∆ ν ) Pr o of. F or an y 1 -Lips hitz funtion ϕ on R n ,     Z ϕ d f # µ − Z ϕ d g # µ     =     Z ϕ ◦ f d µ − Z ϕ ◦ g d µ     6 Lip ϕ Z k f − g k d µ 6 k f − g k L 1 ( µ ) F or the seond inequalit y , let us rst remark that there exists t w o p ositiv e measures µ r and ν r su h that µ = µ ∩ ν + µ r and ν = µ ∩ ν + ν r . Then, d bL ( f # µ, g # ν ) 6 d bL ( f # µ, f # µ ∩ ν ) + d bL ( f # µ ∩ ν , g # µ ∩ ν ) + d bL ( g # µ, g # µ ∩ ν ) No w let us b ound one of the extreme terms of the sum, ∀ ϕ s.t k ϕ k ∞ 6 1 ,     Z ϕ d f # µ − Z ϕ d f # µ ∩ ν     =     Z ϕ ◦ f d µ r     6 mass( µ r ) One onludes using that µ r + ν r = µ ∆ ν . Cor ollar y I.6 . If K and K ′ ar e two  omp at subsets of R n , d bL ( µ K,r , µ K ′ ,r ) 6 k p K − p ′ K k L 1 ( K r ∩ K ′ r ) + H n ( K r ∆ K ′ r ) Hene to get a quan titativ e on tin uit y estimate for the map K 7→ µ K,r one needs to sho w that if K and K ′ are Hausdor-lose, K r ∆ K ′ r is small, and to ev aluate the on tin uit y mo dulus of K 7→ p K ∈ L 1 ( K r ∩ K ′ r ) . This is the purp ose of the t w o follo wing paragraphs. RR n ° 6219 10 Chazal, Cohen-Steiner & Mérigot I I K r ∆ K ′ r is small when K and K ′ are lose It is not hard to see that if d H ( K, K ′ ) is smaller than ε , then K r ∆ K ′ r is on tained in ( K r + ε \ K r − ε ) . The v olume of this thi k tub e around K an then b e expressed as an in tegral of the area of the h yp ersurfaes ∂ K t . The next prop osition giv es a b ound for the measure of the r -lev el set ∂ K r of a ompat set K ⊆ R n dep ending only on its o v ering n um b er N ( K , r ) ( ie. the minimal n um b er of losed balls of radius r needed to o v er K ). In what follo ws, K r is the set of p oin ts of R n at distane less than r of K , and ∂ K r is the b oundary of this set, ie. the r -lev el set of d K . In this paragraph, w e pro v e the follo wing theorem : Theorem. If K is a  omp at set of R n , for every p ositive r , ∂ K r is H n − 1 r e tiable and H n − 1 ( ∂ K r ) 6 N ( ∂ K, r ) × ω n − 1 (2 r ) This prop osition impro v es o v er a result of niteness of the lev el sets of the distane funtion to a ompat set, pro v ed b y b y Oleksiv and P esin in [OP85 ℄. W e b egin b y pro ving it in the sp eial ase of  r -o w ers. A r -o w er F is the the b oundary of the r -tub e of a ompat set on tained in a ball B ( x, r ) , ie. F = ∂ K r where K ⊆ B ( x, r ) . The dierene with the general ase is that if K ⊆ B ( x, r ) , then K r is a star-shap ed set with resp et to x . Th us w e an dene a ra y-sho oting appliation s K : S n − 1 → ∂ K r whi h maps an y v ∈ S n − 1 to the in tersetion of the ra y emanating from x with diretion v with ∂ K r . x v s K ( v ) Figure I I.2: Ra y-sho oting from the en ter of a o w er. Lemma I I.1 . Let K = { e } ⊆ B ( x, r ) and dene s e as ab o v e. Then s e is 2 r -Lips hitz (with resp et to the sphere's inner metri) and its Jaobian is at most (2 r ) n − 1 . Pr o of. Solving the equation k x + tv − e k = r with t > 0 giv es s e ( v ) = x +  q h v | x − e i 2 + r 2 − k x − e k 2 − h v | x − e i  v Denote b y H v the orthogonal of the 2 -plane P spanned b y v and s e ( v ) − e . F or ea h v etor w  hosen in H v , a simple alulation giv es: s e ( v + tw ) = s e ( v ) + tw k s e ( v ) − x k + o ( t 2 ) INRIA Boundary me asur es 11 Hene the deriv ativ e of s e along H v is simply the m ultipliation b y k s e ( v ) − x k 6 2 r . No w, w e no w onsider the ase of the 2 -plane P . W e denote b y θ the angle b et w een s e ( v ) − x and s e ( v ) − e and b y w a v etor tangen t to v in the in tersetion of the sphere with P . Then k (d s e ) v ( w ) k k w k = k s e ( v ) − x k | cos( θ ) | No w let us remark that k s e ( v ) − e k k s e ( v ) − x k | cos( θ ) | = |h s e ( v ) − e | s e ( v ) − x i| = 1 2 ( k x − s e ( v ) k 2 + k s e ( v ) − e k 2 − k x − e k 2 ) > 1 2 k x − s e ( v ) k 2 Finally w e ha v e pro v ed that k (d s e ) v k 6 2 r . The result follo ws b y in tegration. W e denote b y ω n ( r ) the n -Hausdor measure of the n -sphere of radius r . Cor ollar y I I.2 . A r -ower in R n is a H n − 1 r e tiable set and its me asur e is at most ω n − 1 (2 r ) . Pr o of. Let K ⊆ B ( x, r ) b e the ompat set generating the o w er ∂ K r . As ab o v e, for an y v etor v ∈ S n − 1 , w e denote b y s the in tersetion of the ra y { x + tv ; t > 0 } with ∂ K r . Sine K r is a star-shap ed set around x , s is a bijetion from S n − 1 to ∂ K r . No w let ( y k ) b e a dense sequene in K , and denote b y s k the pro jetion from S n − 1 to the o w er ∂ ( ∪ i 6 k { y i } ) r dened as ab o v e. Then ( s k ) on v erges simply to p on S n − 1 . Indeed, if w e x v ∈ S n − 1 and ε > 0 , the segmen t joining x and s ( v ) trunated at a distane ε of s ( v ) is a ompat set on tained in int K r . It is o v ered b y the union ∪ i B ( y i , r ) , so that for N big enough it is also o v ered b y ∪ k 6 N B ( y k , r ) . F or those N , k s k ( x ) − s ( x ) k 6 ε . Finally , ∂ K r is the image of the sphere b y p , whi h is 2 r -Lips hitz as a simple limit of 2 r -Lips hitz funtions. W e no w dedue a general b ound on the measure of the tub e b oundary ∂ K r around a general ompat set K b y o v ering it with a family of o w ers: Theorem I I.3 . If K is a  omp at set of R n , for every p ositive r , ∂ K r is a H n − 1 -r e tiable subset of R n and mor e over, H n − 1 ( ∂ K r ) 6 N ( ∂ K, r ) × ω n − 1 (2 r ) Pr o of. It is easy to see that ∂ K r ⊆ ∂ ( ∂ K r ) . Th us, if w e let ( x i ) b e an optimal o v ering of ∂ K b y op en balls of radius r , and denote b y K i the (ompat) in tersetion of ∂ K with B ( x i , r ) , the b oundary ∂ K r is on tained in the union ∪ i ∂ K r i . Hene its Hausdor measure do es not exeed the sum P i H n − 1 ( ∂ K r i ) . One onludes b y applying the preeding lemma. RR n ° 6219 12 Chazal, Cohen-Steiner & Mérigot R emark I I.4 . 1. The b ound in the theorem is tigh t, as one an  he k taking K = B (0 , r ) . 2. Let us notie that for some onstan t C ( n ) , N ( B (0 , 1 ) , r ) 6 1 + C ( n ) r − n . F rom this and the ab o v e b ound it follo ws that H n − 1 ( ∂ K r ) 6 (1 + C ( n ) × (diam( K ) /r ) n ) ω n − 1 (2 r ) 6 C ′ ( n ) × (1 + diam( K ) n r ) for some univ ersal onstan t C ′ ( n ) dep ending only on the am bien t dimension n . This last inequalit y w as the one pro v ed in [OP85 ℄. T o onlude w e use a w eak form ulation of the  o-ar e a formula , a standard result of geometri measure theory ([DG54℄, [F ed59 ℄), whi h reads Z R n |∇ x f | d H n ( x ) = Z R H n − 1 ( f − 1 ( y ))d H 1 ( y ) whenev er f : R n → R is a Lips hitz map. F rom this form ula and the previous estimation follo ws that Cor ollar y I I.5 . F or any  omp at sets K, K ′ ⊆ R n , with d H ( K, K ′ ) 6 ε , H n ( K r ∆ K ′ r ) 6 Z r + ε r − ε H n − 1 ( ∂ K t )d t 6 2 N ( K, r − ε ) ω n − 1 (2 r + 2 ε ) × ε I I I The map K 7→ p K is lo ally 1 / 2 -Hölder W e no w study the on tin uit y mo dulus of the map K 7→ p K ∈ L 1 ( E ) , where E is a suitable op en set. W e remind the reader of t w o w ell-kno wn fats of on v ex analysis (see for instane [Cla83 ℄): 1. If f : Ω ⊆ R n → R is a lo ally on v ex funtion, its sub dieren tial at a p oin t x , denoted b y ∂ x f is the set of v etors v of R n su h that for all h ∈ R n small enough, f ( x + h ) > f ( x ) + h h | v i . Then f admits a deriv ativ e at x i ∂ x f = { v } is a singleton, in whi h ase ∇ x f = v . 2. A lo ally on v ex funtion has a deriv ativ e almost ev erywhere. Lemma I I I.1 . The funtion v K : R n → R , x 7→ k x k 2 − d K ( x ) 2 is on v ex with gradien t ∇ v K = 2 p K almost ev erywhere. Pr o of. By denition, v K ( x ) = sup y ∈ K k x k 2 − k x − y k 2 = sup y ∈ K v K,y ( x ) with v K,y ( x ) = 2 h x | y i − k y k 2 . Hene v K is on v ex as a suprem um of ane funtions. Beause v K,p K ( x ) and v K tak e the same v alue at x , ∂ x v K,p K ( x ) = { 2 p K ( x ) } ⊆ ∂ v K . Sine v K is dieren tiable almost ev erywhere, equalit y m ust b e true almost ev erywhere whi h onludes the pro of. INRIA Boundary me asur es 13 This lemma sho ws that k p K − p K ′ k L 1 ( E ) = 1 / 2 k∇ v K − ∇ v K ′ k L 1 ( E ) . Our estimation of the on tin uit y mo dulus of the map K 7→ p K will follo w from a general theorem whi h asserts that if ϕ and ψ are t w o uniformly lose on v ex funtions with b ounded gradien ts then ∇ ϕ and ∇ ψ are L 1 -lose. The next prop osition b elo w is the 1 -dimensional v ersion of this result, from whi h w e then dedue the general theorem. Pr oposition I I I.2 . If I is an interval, and ϕ : I → R and ψ : I → R ar e two  onvex funtions suh that diam( ϕ ′ ( I ) ∪ ψ ′ ( I )) 6 k , then letting δ = k ϕ − ψ k L ∞ ( I ) , Z I | ϕ ′ − ψ ′ | 6 6 π (leng th( I ) + k + δ 1 / 2 ) δ 1 / 2 Lemma I I I.3 . Let f : I → R b e a nondereasing funtion with diam ϕ ( I ) 6 k . Then, if F is the ompleted graph of f , ie. the set of p oin ts ( x, y ) ∈ I × R su h that lim x − ϕ 6 y 6 lim x + ϕ , then H n ( F r ) 6 3 π (length( I ) + k + r ) × r . Pr o of. Let γ : [0 , 1 ] → F b e a on tin uous parametrization of F , inreasing with resp et to the lexiographi order on R 2 . Then, for an y inreasing sequene ( t i ) ∈ [0 , 1] and ( x i , y i ) = γ ( t i ) , X i k γ ( t i +1 ) − γ ( t i ) k 6 X i x i +1 − x i + y i +1 − y i 6 length( I ) + k Hene length( F ) 6 length( I ) + k . Th us w e an  ho ose a 1 -Lips hitz parametrization of F , ˜ γ : [0 , length( I ) + k ] → F . Then for an y p ositiv e r , the set X = { ˜ γ ( i × r ) ; 0 6 i 6 N } with N the upp er in teger part of (length( I ) + k ) /r , is su h that an y p oin t of F is at distane at most r of X . Hene F r is on tained in X 2 r , implying that H n ( F r ) 6 N π (3 r / 2) 2 6 3 π (length( I ) + k + r ) r . Pr o of of pr op osition III.2. Let I = [ a, b ] and J = [ c, c + k ] b e su h that ϕ ′ ( I ) ∪ ψ ′ ( I ) ⊆ J . Without loss of generalit y w e will supp ose that ψ ′ ( a ) = ϕ ′ ( a ) = c and ψ ′ ( b ) = ϕ ′ ( b ) = c + k . With this assumption, the ompleted graphs Φ and Ψ of ϕ ′ and ψ ′ dened as ab o v e are t w o retiable urv es joining ( a, c ) and ( b, c + k ) . W e let V b e the set of p oin ts ( x, y ) ∈ R 2 lying b et w een those graphs; the quan tit y w e w an t to b ound is R I | ϕ ′ − ψ ′ | = H 2 ( V ) . Let δ = k ϕ − ψ k L ∞ ( I ) . F or an y p oin t p = ( x, y ) in V , and an y δ ′ > δ , the losed disk D = B ( p, p 2 δ ′ /π ) of v olume 2 δ ′ en tered at p annot b e on tained in V . Indeed if it w ere, then the dierene κ = ϕ − ψ w ould inrease to o m u h around p : sine κ ′ has a onstan t sign on this segmen t, | κ ( x + 2 δ ′ /π ) − κ ( x − 2 δ ′ /π ) | = Z x +2 δ ′ /π x − 2 δ ′ /π | κ ′ | > H 2 ( D ) = 2 δ ′ > 2 δ This on tradits k κ k ∞ = δ . Hene, D m ust in tersets ∂ V implying that V m ust b e on tained in ( ∂ V ) √ 2 δ ′ /π for an y δ ′ > δ . Sine ∂ V = Φ ∪ Ψ , the previous lemma giv es H 2 ( V ) 6 H 2  Φ √ 2 δ ′ /π  + H 2  Ψ √ 2 δ ′ /π  6 6 π (length( I ) + k + p 2 δ ′ /π ) p 2 δ ′ /π Letting δ ′ on v erge to δ onludes the pro of. RR n ° 6219 14 Chazal, Cohen-Steiner & Mérigot A generalization of this prop osition in arbitrary dimension will follo w from an argumen t oming from in tegral geometry , ie. w e will in tegrate the inequalit y of prop osition I I I.2 o v er the set of lines of R n to get a b ound on k∇ ϕ − ∇ ψ k L 1 ( E ) . W e let L n b e the set of orien ted ane lines in R n seen as the submanifold of R 2 n made of p oin ts ( u, p ) ∈ R n × R n with u ∈ S n − 1 and x in the h yp erplane { u } ⊥ , and endo w ed with the indued Riemannian metri. The orresp onding measure d L n is in v arian t under rigid motions. W e let D n u b e the set of orien ted lines with a xed diretion u . The usual Crofton form ula ( f. [Mor88 ℄ for instane) states that for an y H n − 1 retiable subset S of R n , with β n the v olume of the unit n -ball, H n − 1 ( S ) = 1 2 β n − 1 Z ℓ ∈L n #( ℓ ∩ S )d ℓ (I I I.1) where # X is the ardinalit y of X . W e will also use the follo wing Crofton-lik e form ula: if K is a H n retiable subset of R n , H n ( K ) = 1 ω n − 1 Z ℓ ∈L n H 1 ( ℓ ∩ K )d ℓ (I I I.2) whi h follo ws from the F ubini theorem (remem b er ω n − 1 is the v olume of the ( n − 1) sphere). Lemma I I I.4 . Let X : E → R n b e a L 1 -v etor eld on an op en subset E ⊆ R n . Z E k X k = n 2 ω n − 2 Z ℓ ∈L n Z y ∈ ℓ ∩ E |h X ( y ) | u ( ℓ ) i| d y d ℓ Sketh of pr o of. The family of v etor elds of the form P i X i χ Ω i , where the Ω i are a nite n um b er of disjoin t op en subsets of R n and X i are onstan t v etors, is L 1 -dense in the spae L 1 ( R n , R n ) . Using this fat and the on tin uit y of the t w o sides of the equalit y , it is enough to pro v e this equalit y for X = x k X k χ E where x is a onstan t unit v etor and E a b ounded op en set of R n . In that ase, one has Z ℓ ∈D n u Z y ∈ ℓ |h X ( y ) | u i| d y d ℓ = k X k |h x | u i| Z ℓ ∈D n u length( E ∩ ℓ )d ℓ = k X k L 1 ( E ) |h x | u i| By a F ubini-lik e theorem one has Z ℓ ∈L n Z y ∈ ℓ |h X ( y ) | u ( ℓ ) i| d y d ℓ = Z u ∈S n − 1 Z ℓ ∈D n u Z y ∈ ℓ |h X ( y ) | u ( ℓ ) i| d y d ℓ d u = k X k L 1 ( E ) Z u ∈S n − 1 |h x | u i| d u INRIA Boundary me asur es 15 The last in tegral do es, in fat, not dep end on x and its v alue an b e easily omputed: Z u ∈S n − 1 |h x | u i| d u = 2 ω n − 2 Z 1 0 t (1 − t 2 ) n 2 − 1 d t = 2 n ω n − 2 Theorem I I I.5 . L et E b e an op en subset of R n with ( n − 1) r e tiable b oundary, and f , g b e two lo  al ly  onvex funtions on E suh that diam( ∇ f ( E ) ∪ ∇ g ( E )) 6 k . Then, letting δ = k f − g k L ∞ ( E ) k∇ f − ∇ g k L 1 ( E ) 6 C 1 ( n )( H n ( E ) + ( k + δ 1 / 2 ) H n − 1 ( ∂ E )) δ 1 / 2 with C 1 ( n ) 6 6 π n as so on as n > 5 (in fat, C 1 ( n ) = O ( √ n ) ). Pr o of of the the or em. The 1 -dimensional ase follo ws from prop osition I I I.2: in that ase, E is a oun table union of in terv als on whi h f and g satisfy exatly the h yp othesis of the prop osition. Summing the inequalities giv es the result with C 1 (1) = 6 π . The general ase will follo w from this one with the use of in tegral geometry . If w e set X = ∇ f − ∇ g , f ℓ = f | ℓ ∩ E and g ℓ = g | ℓ ∩ E . Lemma I I I.4 giv es, letting D ( n ) = n/ (2 ω n − 2 ) , Z E k∇ f − ∇ g k = D ( n ) Z ℓ ∈L n Z y ∈ ℓ ∩ E |h∇ f − ∇ g | u ( ℓ ) i| d y d ℓ = D ( n ) Z ℓ ∈L n Z y ∈ ℓ ∩ E | f ′ ℓ − g ′ ℓ | d y d ℓ The funtions f ℓ and g ℓ satisfy the h yp othesis of the one-dimensional ase, so that for ea h  hoie of ℓ , and with δ = k f − g k L ∞ ( E ) , Z y ∈ ℓ ∩ E | f ′ ℓ − g ′ ℓ | d y 6 6 π D ( n )( H 1 ( E ∩ ℓ ) + ( k + δ 1 / 2 ) H 0 ( ∂ E ∩ ℓ )) δ 1 / 2 It follo ws b y in tegration on L n that Z E k∇ f − ∇ g k 6 6 π D ( n )  Z L n H 1 ( E ∩ ℓ )d L n + ( k + δ 1 / 2 ) Z L n H 0 ( ∂ E ∩ ℓ )d L n  δ 1 / 2 The form ula I I I.1 and I I I.2 sho w that the rst in tegral is equal (up to a onstan t) to the v olume of E and the seond to the ( n − 1) -measure of ∂ E . This pro v es the theorem with C 1 ( n ) = 6 π D ( n )( ω n − 1 + 2 β n − 1 ) . T o get the b ound on C 1 ( n ) one uses the form ula ω n − 1 = nβ n and β n +1 6 β n as so on as n > 5 . Multiplying f and g b y the same p ositiv e fator t and optimizing the result in t yields a b etter, homogeneous, b ound : RR n ° 6219 16 Chazal, Cohen-Steiner & Mérigot Cor ollar y I I I.6 . Under the same hyp othesis as in the or em III.5, one gets the fol lowing b ound, with δ = k f − g k L ∞ ( E ) : k∇ f − ∇ g k L 1 ( E ) 6 2 C 1 ( n )[( H n ( E ) H n − 1 ( ∂ E ) diam( ∇ f ( E ) ∪ ∇ g ( E ))) 1 / 2 + H n − 1 ( ∂ E ) δ 1 / 2 ] δ 1 / 2 R emark I I I.7 . T o get an homogeneous b ound as in this orollary , one ould also optimize the one-dimensional b ound of prop osition I I I.2 b efore in tegrating on the set of ane lines of R n as in the pro of of theorem I I I.5. The b ound obtained this w a y is alw a ys stritly b etter than the ones of b oth theorem I I I.5 and orollary I I I.6, but in v olv es an in tegral term Z ℓ ∈L n p H 0 ( ℓ ∩ ∂ E ) H 1 ( ℓ ∩ E )d ℓ whose in tuitiv e meaning is not quite lear. Applying theorem I I I.5 to the funtions v K and v K ′ in tro dued at the b egining of this part and using lemma I I I.1, one easily gets : Cor ollar y I I I.8 . If E is an op en set of R n with r e tiable b oundary, K and K ′ two  omp at subsets of R n then, with R K = k d K k L ∞ ( E ) and ε = d H ( K, K ′ ) , k p K − p K ′ k L 1 ( E ) 6 C 1 ( n )[ H n ( E ) + (diam( K ) + ε + (2 R K + ε ) 1 / 2 ε 1 / 2 ) H n − 1 ( ∂ E )] × (2 R K + ε ) 1 / 2 ε 1 / 2 In p artiular, if d H ( K, K ′ ) is smal ler than min( R K , diam( K ) , diam( K ) 2 /R K ) , ther e is an- other  onstant C 2 ( n ) dep ending only on n suh that k p K − p K ′ k L 1 ( E ) 6 C 2 ( n )[ H n ( E ) + diam( K ) H n − 1 ( ∂ E )] p R K d H ( K, K ′ ) R emarks I I I.9 . 1. This theorem giv es in partiular a quan titativ e v ersion of the on ti- n uit y theorem 4 . 13 of [ F ed59 ] : if ( K n ) is a sequene of ompat subsets of R n with reach( K n ) > r > 0 , on v erging to a ompat set K , then reach( K ) > r and p K n on v erges to p K uniformly on ea h ompat set on tained in { x ∈ R n ; d K ( x ) < r } . Ho w ev er w e ha v e to stress that the result w e ha v e pro v ed is more general sine it do es not mak e an y assumption on the regularit y of K n  at the exp ense of uniform on v ergene. 2. The seond term of the b ound in v olving H n − 1 ( ∂ E ) is neessary . Indeed, let us supp ose that a b ound k p K − p K ′ k L 1 ( E ) 6 C ( K ) H n ( E ) √ ε w ere true around K for an y op en set E . No w let K b e the union of t w o parallel h yp erplane at distane R in terseted with a big sphere en tered at a p oin t x of their medial h yp erplane M . Let E ε b e a ball of radius ε tangen t to M at x and K ε b e the translation b y ε of K along the ommon normal of the h yp erplanes su h that the medial h yp erplane of K ε tou hes the ball E ε on the opp osite of x . Then, for ε small enough, k p K − p K ′ k L 1 ( E ε ) ≃ R × H n ( E ε ) , whi h learly exeeds the assumed b ound for a small enough ε . INRIA Boundary me asur es 17 3. A ording to this theorem, the map K 7→ p K ∈ L 1 ( E ) is lo ally 1 / 2 -Hölder. The follo wing example sho ws that this result annot b e impro v ed ev en around a v ery simple ompat set. ℓ R Figure I I I.3: A sequene of knife blades on v erging to a segmen t. Let S and S ′ b e t w o opp osite sides of a retangle E , ie. t w o segmen ts of length L and at distane R . W e no w dene a Hausdor appro ximation of S : for an y p ositiv e in teger N , divide S in N small segmen ts s i of ommon length ℓ , and let C i b e the unique irle with en ter in S ′ whi h on tains the t w o endp oin ts of s i . W e no w let S N b e the union of the irle ars of C i omprised b et w een the t w o endp oin ts of s i . Then it is not v ery hard to see that if R ε = R + ε is the ommon radius of all the C i , R 2 ε = R 2 + ( ℓ/ 2) 2 , ie. d H ( S, S N ) = p R 2 + ( ℓ/ 2) 2 − R 6 R ℓ 2 / 8 . Then the L 1 -distane b et w een the pro jetions on S and S N is at least Ω( ℓ ) (b eause almost half of the p oin ts in E pro jets on the orners of S N , see the shaded area in g. I I I.3). Hene, k p S − p S N k L 1 ( E ) = Ω( ℓ ) = Ω(d H ( S, S N ) 1 / 2 ) Replaing L 1 ( E ) with L 1 ( µ ) where µ has b ounded v ariation As w e ha v e seen b efore, a orollary of the previous result is that if µ = H n | E , the map K 7→ p K # µ is lo ally 1 / 2 -Hölder. This result an b e generalized when µ = u H n where u ∈ L 1 lo c ( R n ) has b ounde d variation . W e reall some fats ab out the theory of funtions with b ounded v ariation, tak en from [AFP00℄. If Ω ⊆ R n is an op en set and u ∈ L 1 lo c (Ω) , the variation of u in Ω is V( u, Ω ) = sup  Z Ω u div ϕ ; ϕ ∈ C 1 c (Ω) , k ϕ k ∞ 6 1  A funtion u ∈ L 1 lo c (Ω) has b ounde d variation if V ( u, Ω) < + ∞ . The set of funtions of b ounded v ariation on Ω is denoted b y BV(Ω) . W e also men tion that if u is Lips hitz on Ω , then V( u, Ω ) = k∇ u k L 1 (Ω) . Finally , w e let V( u ) b e the total v ariation of u in R n . Theorem I I I.10 . L et µ ∈ M ( R n ) b e a me asur e with density u ∈ BV( R n ) with r esp e t to the L eb esgue me asur e, and K b e a  omp at subset of R n . W e supp ose that supp( u ) ⊆ K R . Then, if d H ( K, K ′ ) is smal l enough, d bL (p K # µ, p K ′ # µ ) 6 C 2 ( n )  k u k L 1 ( K R ) + diam( K ) V ( u )  √ R × d H ( K, K ′ ) 1 / 2 RR n ° 6219 18 Chazal, Cohen-Steiner & Mérigot Pr o of. W e b egin with the additional assumption that u has lass C ∞ . The funtion u an b e written as an in tegral o v er t ∈ R of the  harateristi funtions of its sup erlev el sets E t = { u > t } , ie. u ( x ) = R ∞ 0 χ E t ( x )d t . F ubini's theorem then ensures that for an y Lips hitz funtion f dened on R n with k f k Lip 6 1 , p K ′ # µ ( f ) = Z R n f ◦ p K ′ ( x ) u ( x )d x = Z R Z R n f ◦ p K ′ ( x ) χ { u > t } ( x )d x d t By Sard's theorem, for almost an y t , ∂ E t = u − 1 ( t ) is a ( n − 1) -retiable subset of R n . Th us, for those t the previous orollary implies, for ε = d H ( K, K ′ ) 6 ε 0 = min( R, diam( K ) , diam( K ) 2 /R K ) , Z E t | f ◦ p K ( x ) − f ◦ p K ′ ( x ) | d x 6 k p K − p K ′ k L 1 ( E t ) 6 C 2 ( n )[ H n ( E t ) + diam( K ) H n − 1 ( ∂ E t )] √ Rε Putting this inequalit y in to the last equalit y giv es | p K # µ ( f ) − p K ′ # µ ( f ) | 6 C 2 ( n )  Z R H n ( E t ) + diam( K ) H n − 1 ( ∂ E t )d t  √ Rε Using F ubini's theorem again and the oarea form ula one nally gets that | p K # µ ( f ) − p K ′ # µ ( f ) | 6 C 2 ( n )  k u k L 1 ( K R ) + diam( K ) V ( u )  √ Rε. This pro v es the theorem in the ase of Lips hitz funtions. T o onlude the pro of in the general ase, one has to appro ximate the b ounded v ariation funtion u b y a sequene of C ∞ funtions ( u n ) su h that b oth k u − u n k L 1 ( K R ) and | V( u ) − V( u n ) | on v erge to zero, whi h is p ossible b y theorem 3.9 in [AFP00℄. R emark I I I.11 . T aking u = χ E where E is a suitable op en set sho ws that theorem I I I.8 an also b e reo v ered from I I I.10. IV Stabilit y of b oundary and urv ature measures W e om bine the results of orollaries I.6, I I.5 and I I I.8 to get Theorem IV.1 . If K and K ′ ar e two  omp at sets with ε = d H ( K, K ′ ) smal ler than min(diam K , r, r 2 / diam K ) , then d bL ( µ K,r , µ K ′ ,r ) 6 C 3 ( n ) N ( K , r − ε ) r n [ r + diam( K )] r ε r In p artiular, if for a given b ounde d Lipshitz funtion f on R n , one denes ϕ K,f ( r ) = µ K,r ( f ) , the map K 7→ ϕ K,f ∈ C 0 ([ r min , r max ]) with 0 < r min < r max is lo  al ly 1 / 2 -Hölder. INRIA Boundary me asur es 19 In what follo ws w e supp ose that ( r i ) is a sequene of n distint n um b ers 0 < r 0 < ... < r n . F or an y ompat set K and f ∈ C 0 ( R n ) , w e let h Φ ( r ) K,i ( f ) i i b e the solutions of the linear system ∀ i s.t 0 6 i 6 n , n X j =0 ω n − j Φ ( r ) K,j ( f ) r n − j i = µ K,r i ( f ) Sine the system is linear in ( µ K,r i ( f )) and these v alues dep ends on tin uously on f , the map f 7→ Φ ( r ) K,i ( f ) is also linear and on tin uous, ie. Φ ( r ) K,i is a signed measure on R n . It is also to b e notied that if K has p ositiv e rea h with reach ( K ) > r n , the Φ ( r ) K,i oinide with the usual urv ature measures of K . In that ase, the follo wing result giv es a w a y to appro ximate the (usual) urv ature measures of K from a Hausdor-appro ximation of it ev en if its rea h is arbitrary small. Cor ollar y IV.2 . Ther e exist a  onstant C dep ending on K and ( r ) suh that for any  omp at subset K ′ of R n lose enough to K , ∀ i, d bL  Φ ( r ) K ′ ,i , Φ ( r ) K,i  6 C d H ( K, K ′ ) 1 / 2 Referenes [AFP00℄ L. Am brosio, N. F uso, and D. P allara. F untions of b ounde d variation and fr e e dis ontinuity pr oblems . Oxford Mathematial Monographs, 2000. [BGV07℄ F. Bolley , A. Guillin, and C. Villani. Quan titativ e Conen tration Inequalities for Empirial Measures on Non-ompat Spaes. Pr ob ability The ory and R elate d Fields , 137(3):541593, 2007. [BKL06℄ A. Beygelzimer, S. Kak ade, and J. Langford. Co v er trees for nearest neigh b or. Pr o  e e dings of the 23r d international  onfer en e on Mahine le arning , pages 97 104, 2006. [BSK℄ G.H. Bendels, R. S hnab el, and R. Klein. Deteting Holes in P oin t Set Surfaes. [Cla83℄ F.H. Clark e. Optimization and nonsmo oth analysis . Wiley New Y ork, 1983. [DG54℄ E. De Giorgi. Su una teoria generale della misura (r- 1)-dimensionale in uno spazio adr dimensioni. A nnali di Matemati a Pur a e d Appli ata , 36(1):191213, 1954. [DGGZ03℄ T.K. Dey , J. Giesen, S. Gosw ami, and W. Zhao. Shap e Dimension and Appro x- imation from Samples. Disr ete and Computational Ge ometry , 29(3):419434, 2003. RR n ° 6219 20 Chazal, Cohen-Steiner & Mérigot [DHOS07℄ J. Daniels, L. K. Ha, T. O hotta, and C. T. Silv a. Robust smo oth feature ex- tration from p oin t louds. In Shap e Mo deling International, 2007. Pr o  e e dings , 2007. [F ed59℄ H. F ederer. Curv ature Measures. T r ansations of the A meri an Mathemati al So iety , 93(3):418491, 1959. [FK06℄ S. F unk e and C. Klein. Hole detetion or: ho w m u h geometry hides in onne- tivit y? Pr o  e e dings of the twenty-se  ond annual symp osium on Computational ge ometry , pages 377385, 2006. [GWM01℄ S. Gumhold, X. W ang, and R. MaLeo d. F eature extration from p oin t louds. Pr o . 10th International Meshing R oundtable , pages 293305, 2001. [Mor88℄ F. Morgan. Ge ometri Me asur e The ory: A Be ginner's Guide . A ademi Press, 1988. [OP85℄ I.Y. Oleksiv and NI P esin. Finiteness of Hausdor measure of lev el sets of b ounded subsets of Eulidean spae. Mathemati al Notes , 37(3):237242, 1985. [RBBK06℄ G. Rosman, A. M. Bronstein, M. M. Bronstein, and R. Kimmel. T op ologially onstrained isometri em b edding. In Pr o . Conf. on Mahine L e arning and Pat- tern R e  o gnition (MLPR) , 2006. [Vil03℄ C. Villani. T opis in Optimal T r ansp ortation . Amerian Mathematial So iet y , 2003. [W ey39℄ H. W eyl. On the V olume of T ub es. A meri an Journal of Mathematis , 61(2):461 472, 1939. 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