Measure-Theoretic Properties of Level Sets of Distance Functions

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1 Measure-Theorei Properies of Level Ses of Disae Fuios Daiel Kraf Uiversiy of Graz Isiue of Mahemais, NAWI Graz Uiversiäsplaz 3, 8010 Graz, Ausria July 22, 2015 Absra We osider he level ses of disae fuios from he poi of view of geomeri measure heory. This lays he foudaio for furher researh ha a be applied, amog oher uses, o he derivaio of a shape alulus based o he level-se mehod. Pariular fous is pu o he 1)-dimesioal Hausdorff measure of hese level ses. We show ha, sarig from a bouded se, all sub-level ses of is disae fuio have fiie perimeer. Furhermore, if a uiform-desiy odiio is saisfied for he iiial se, oe a eve show a upper boud for he perimeer ha is uiform for all level ses. Our resuls are similar o exisig resuls i he lieraure, wih he impora disiio ha hey hold for all level ses ad o jus almos all. We also prese a example demosraig ha our resuls are sharp i he sese ha o uiform upper boud a exis if our uiform-desiy odiio is o saisfied. This is eve rue if he iiial se is oherwise very regular i. e., a bouded Caioppoli se wih smooh boudary). Keywords: Geomeri Measure Theory, Level Se, Disae Fuio, Hausdorff Measure, Perimeer, Caioppoli Se 2010 Mahemais Subje Classifiaio: 28A75, 49Q10, 49Q12 Published by Spriger i Joural of Geomeri Aalysis, DOI /s The fial publiaio is available a hp://lik.spriger.om/arile/ /s Iroduio Regulariy of level ses is a wide ad ieresig field. This a be see already i he oex of he lassial Sard heorem ad he o-area formula see page 112 of [11]). For ree work i his direio, le us refer o [1]. I a more speifi oex, we are ieresed i he 1)-dimesioal Hausdorff measure of he level ses of disae fuios: For some ope se Ω 0 R, he disae fuio is give by d Ω0 x) = if y Ω 0 x y. This is a widely sudied osru wih well-kow properies. See, for isae, Chaper 6 of [9]. I pariular, oe ha d Ω0 x) is well-defied ad o-egaive for all x R. Furhermore, he fuio d Ω0 is oiuous o R ad d Ω0 x) = 0 for all x Ω 0. For > 0, le us also defie he level ses Ω = d Ω0 1, )) = {x R d Ω0 x) < } ad Γ = d Ω0 1 {}) = {x R d Ω0 x) = }. 1) Coiuiy of he disae fuio implies ha Ω is ope ad Γ losed. I is also easy o see ha Γ oiides wih he opologial boudary of Ω, i. e., Γ = Ω. The se Ω is someimes alled he -evelope of Ω 0 i he lieraure. I is a iflaed ad smoohed versio of Ω 0. The sudy of he surfae measure H 1 Γ ), whih orrespods roughly o he perimeer P Ω ), is aurally oeed o surfae flows. This siuaio was sudied by Caraballo i [7] ad a be direly moivaed by he famous paper [2] of Almgre, Taylor ad Wag. 1

2 Our mai resuls are very similar, bu hey differ i oe impora aspe: They do o deped o he o-area formula. Cosequely, hey hold for all level ses ad o jus almos all. To highligh a pariular use of his improveme, le us briefly meio he lassial level-se mehod irodued by Osher ad Sehia i [15]: Based o his mehod, oe a desribe evolvig shapes as he sub-zero level ses of a ime-depede level-se fuio. If he geomery is haged by movig he boudary i he ormal direio aordig o a give speed field, he ime evoluio of he level-se fuio a be desribed by he so-alled level-se equaio. Based o his mehod, oe a, for isae, build a framework for shape opimisaio as doe i [5]. See [10], [6] ad [14] for some ree appliaios. Our work i [13] allows us o represe he propagaig domais i he level-se framework wih a formula similar o 1). I he geeral ase, oe uses he soluio of a Eikoal equaio isead of he disae fuio d Ω0. The siuaio osidered here is a speial ase, whih resuls if he speed field is posiive ad osa hroughou all of R. I his framework, oe a also perform shape-sesiiviy aalysis. The resulig shape derivaives are direly oeed o he perimeer of Ω ad, a-priori, defied oly for almos all 0. For he aalysis of opimisaio mehods based o hese derivaives, i is impora o osider he abiliy o oiuously exed he shape derivaives o all imes. Cosequely, i is impora o ask he quesio of oiuiy of he perimeer P Ω ) wih respe o. Oe half of his quesio a be resolved quie easily by he well-kow lower semi-oiuiy propery of he oal variaio. This, i ur, implies lower semi-oiuiy for he perimeer of he evolvig ses. Esimaes i he oher direio, however, are more diffiul o obai. I pariular, we eed upper bouds for P Ω ) ha are o resried by a almos-all qualifiaio i. While his paper a o give a full oiuiy resul, we are, ideed, able o improve upo he exisig resuls i [7] i his direio. Wih he developed ehial ools, i may be possible o prove oiuiy of he perimeer i a fuure work. I Seio 2, we give a example ha demosraes ha blow-up of he perimeer of Ω a happe for 0 + eve if Ω 0 is a smooh Caioppoli se. Afer ivesigaig some auxiliary geomeri properies of spherial seors i Seio 3, we will derive our mai resuls i Seio 4. The firs is a kid of iverse isoperimeri iequaliy see Theorem 3), ha gives a upper boud o he perimeer P Ω ) of Ω i erms of he reaed volume Ω \Ω 0. A obvious esimae of his volume follows if Ω 0 is bouded, whih resuls i Corollary 1. Noe, however, ha his oly yields a upper boud for P Ω ) ha diverges like 1/ for 0 +. This mahes our observaios i Theorem 1. Uder a addiioal uiform-desiy assumpio o Ω 0, we a furher improve he esimae: I his siuaio, he volume a be bouded i erms of he perimeer of he iiial domai Ω 0 imes. Cosequely, we obai a uiform boud o he perimeer of Ω. This will be doe i Subseio 4.2. Subseio 4.3 disusses our uiform-desiy odiio i ompariso o relaed geomeri properies i he lieraure. We will see ha i is srily weaker ha he uiform oe propery, ad a suffiie odiio for he fiie desiy perimeer irodued by Buur ad Zolésio i [4]. Noe ha our mai resuls ad he uiform-desiy assumpio are sharp, as demosraed by he ouerexamples i Seio 2. 2 Moivaig Example for Perimeer Blow-Up Before we sar workig owards he mai resuls, le us give a moivaig example. I shows why i is eessary o irodue he oio of uiform lower desiy i Subseio 4.2 ogeher wih he omplexiies i reaes. There is a lassial exbook example for elemeary geomery: Le a rope be pu ighly aroud he Earh s equaor. If he rope is ow prologed by a sigle mere, how far will i be above he surfae? Wih a rivial alulaio, oe arrives a he surprisig resul ha he disae is o egligible. I fa, he relaioship bewee he hages i a irle s radius ad is perimeer is idepede of he irle s size. We a exploi his fa o jus for huge bu also for iy irles. This allows us o show ha he perimeer of Ω a blow up for 0 + eve if Ω 0 has fiie perimeer ad is bouded: Example 1. Cosider D = [0, 2] [0, 1] R 2 as hold-all domai. For k = 0, 1,..., defie l k = 4 k, r k = l k) 2 4 = k, N k = 2 k l k ) 2 = 8k. Based o hese defiiios, we defie Ω 0 as a ifiie uio of balls as depied i Figure 1. Speifially, Ω 0 is osrued by spliig D firs io a sequee of verial srips wih widhs 2 k. Eah srip is he furher divided io squares of size l k l k. Io eah suh square, we pu a ball wih radius r k. For eah k, here is a oal of N k suh squares ad balls. Eah ball a level k has perimeer 2πr k, so ha he oal perimeer of Ω 0 is give as P Ω 0) = N k 2πr k = π ) k 8 = π Thus, Ω 0 is a bouded se of fiie perimeer. I is also lear ha i has a smooh boudary, sie i osiss eirely of balls. However, sie he radii of he balls beome arbirarily small, he urvaure of Γ 0 is o bouded. 2

3 N 0 = 1 N 1 = 8 r 0 l 0 l 1 l 1 Figure 1: The oaio ad iiial se Ω 0 used i Example 1. Ω 0 osiss of he uio of all grey balls. For he ime evoluio of Ω 0, oe ha eah irle grows ouwards ad is a irle of radius r k + a ime. This works as log as is small eough, so ha he irle does o ye hi aoher growig irle. If we le k be he ime a whih he irles of level k hi heir elosig square, we fid ha k = l k 2 r k = k k = k 12 ) 4 k k. 2) The oher way roud, his meas ha for imes < k, all irles up o ad iludig) level k have eraily o ouhed ay ohers. Le > 0 be give, ad m suh ha m+1 < m. If we use oly irles up o level m o esimae he perimeer of Ω, his yields m m P Ω ) N k 2πr k + ) 2π N k 2π m+1 m 8 k π m+1 1 π 2 m+1 1 ). 3) 4 m Noe ha we a see already here ha his expressio is ubouded for 0 +, sie his limi orrespods o m. To ge a more preise esimae, we a rewrie 2) o ge Combiig his resul wih 3) fially gives 4 m 1 2 m 1 4 m 2 2 m+1 1 m 2 1 m+1 2. P Ω ) π 14 ) 1 2 1, whih diverges like 1/ as 0 + ad eraily beomes ubouded. If oe osiders he alulaios i Example 1 arefully, oe a see ha he base umber i he defiiio of l k four i he example) ifluees oly he osa i fro of he fial esimae as log as i is larger ha wo. The expoe 1/2 deermiig he rae o be 1/ omes from he fa ha eah level of balls ges assiged oly half he area ha was assiged o he previous level. We a irease his fraio as log as i is less ha oe if we sill wa o ge a bouded se as resul. This lie of hough a be exeded o he followig resul: Theorem 1. Le 2 ad 0 < s < 1 be give. There exiss a Caioppoli se Ω 0 R smooh boudary, suh ha bouded ad wih P Ω ) C s for some osa C ad > 0 small eough. I pariular, his rae of divergee holds i he limi 0 +. Proof. We repliae he osruio of Example 1: For he desired resul, hoose some α > 1 ad se f = α s 1 0, 1). Noe ha fα > fα = α s > 1. We defie l k = α k, r k = l k) 4 = α k, N k = f k α k. 4 3

4 This leads o a oal volume of all l k ) -ubes of ) N k l k ) f k α k + 1 α k = f k + α ) k 1 = 1 f + 1 <. 1 α Hee, sie f < 1, we a fi everyhig io a bouded se as before. Clearly, Ω 0 has agai a smooh boudary. Is perimeer is also fiie sie ) P Ω 0) = C N k r k ) 1 C f k α k + 1 r k = C f k + α ) ) k = C 4 4 ) 1 1 f α O he oher had, we sill fid ha balls a level k have o ye hi ayhig else uil ime Thus, for > 0 wih m+1 < m, we kow ha k = l k 2 r k = α k 2 α k α k 4 2 α k = α k ) m m ) P Ω ) C N k r k + ) 1 C m+1) 1 α m+1) 1 m N k C fα ) k 4 = C α 1 ) m+1) f m+1 α ) m fα 1 C 4 1 fα 1) where we have defied he osa C suiably. From 4), i follows ha fα) m+1 1 ) = C α s ) m+1 1), α m m α s ) m+1 4 s s. Combiig his wih he esimae for P Ω ) above shows he laim. 3 Auxiliary Geomeri Resuls I order o show our mai resuls i Seio 4 i pariular, Theorem 3), we eed some auxiliary resuls. They are oly based o elemeary geomery ad will be prepared i his seio. The basi obje sudied is wha we will all a seor below: Defiiio 1. Le x 0, x R ad φ [0, π/2]. We defie S φ x 0, x) = {y R 0 < x 0 y < x 0 x ad y x 0) x x 0) > x 0 y x 0 x os φ }. We will ofe se = x 0 x o be he seor s radius. The se S φ x 0, x) is a ope seor of he ball wih ere x 0 ad radius. The value of φ, whih orrespods o he maximum allowed agle x x 0 y, defies he seor s aperure. Besides usig he agle φ direly, we will also eed o defie suh a seor via a auxiliary ball B δ x) for δ <. The idea is depied i Figure 2b: I his ase, he seor s aperure is defied idirely via δ. I is hose as he agle a whih he ball aroud x ierses he larger sphere wih ere x 0. Wih basi rigoomery, oe a derive ) φδ) = aros 1 δ2 5) 2 2 for he orrespodig aperure agle. I he followig, we will oly eed wo basi properies of his explii fuio: δ < φδ) holds for all δ ad φδ)/δ 1 i he limi δ 0 +. I oher words, φδ) δ asympoially for small δ. The firs par of our geomeri aalysis of seors is oered wih deermiig heir volume i. e., - dimesioal Lebesgue measure). For his, le us sae he followig fudameal geomeri fas: Lemma 1. Le 2. The volume of a ball wih radius ρ > 0 is give by vol B ρ x)) = ω ρ, ω = π /2 Γ/2 + 1). 4

5 x x δ φ x 0 a) Based o φ aordig o Defiiio 1. x 0 b) Defied via he ball B δ x) ad 5). Figure 2: Defiiios of he seor S φ x 0, x). This holds obviously for arbirary x R. Furhermore, here exiss a mappig r : [0, π/2] [0, 1/2] whih is oiuous, bijeive, srily ireasig ad saisfies vol S φ x 0, x)) = rφ) vol B 0)) = rφ) ω 6) for all x 0, x R ad φ [0, π/2]. Here, we have se = x 0 x as before. I addiio, exiss ad is srily posiive. rφ) lim > 0 7) φ 0 + φ 1 Proof. The volume of -dimesioal balls is a well-kow resul. See, for isae, Theorem i [16]. The remaiig saemes follow by a rouie alulaio i spherial oordiaes. We a also relae he surfae area of a seor s base o is volume. This resul will be used laer whe we prove Theorem 3. I follows immediaely from Lemma 1 ad, i pariular, 7): Lemma 2. For fixed > 0, here exis δ 0 > 0 ad a dimesioal osa C suh ha δ 1 ω 1 C vol S φδ) x 0, x) ) for all δ 0, δ 0) ad arbirary x 0, x R wih x 0 x =. Fially, le us osider wo seors S φ x 0, x) ad S φ y 0, y). The agle is he same for boh, ad we assume ha φ = φδ) for some δ > 0. Le also = x 0 x = y 0 y. We are pariularly ieresed i he siuaio B δ x) B δ y) = ad mi x 0 y, y 0 x ). 8) A mai igredie for he proof of Theorem 3 is he fa ha his odiio is suffiie for boh seors o be disjoi. This is illusraed i Figure 3: If he balls are disjoi, we a osru he hyperplae H ha divides he lie x y a is midpoi ad is perpediular o i. This plae has he propery ha all pois above i are loser o x ha o y, ad vie-versa for pois o he oher side. Thus, 8) implies ha x 0 is o he same side as x, while y 0 mus be o he oher side ogeher wih y. Hee, he plae separaes he ovex ses S φ x 0, x) ad S φ y 0, y) from eah oher, whih meas ha he seors mus be disjoi. This is he mai idea behid he followig resul: Lemma 3. Le δ > 0 ad x 0, y 0, x, y R wih = x 0 x = y 0 y suh ha 8) holds. The S φδ) x 0, x) S φδ) y 0, y) =. 5

6 x x 0 H y 0 y Figure 3: The hyperplae H separaes boh he balls ad he full seors from eah oher whe 8) holds. This is he mai idea i he proof of Lemma 3. Proof. Wih a proper raslaio, we a assume, wihou loss of geeraliy, ha y = x. Beause B δ x) ad B δ y) are disjoi, he hyperplae H = {p R x p = 0} separaes boh balls see Figure 3). Furhermore, by 8) we kow x 0 x 2 = 2 x 0 y 2 = x 0 + x 2. Muliplyig his iequaliy ou, we fid 0 x x 0. This meas ha x ad x 0 are o he same side of H. Similarly, we also fid ha y ad y 0 are o oe side of H. Sie y = x, his meas 0 y y 0 x y 0 0. Hee, x 0 ad y 0 are o differe sides of he hyperplae H. Thus, H separaes also he ovex hulls of B δ x) {x 0} ad B δ y) {y 0}, whih oai S φ x 0, x) ad S φ y 0, y), respeively. This shows ha he seors are, ideed, disjoi. 4 Mai Resuls Wih he preparaios of Seio 3 i plae, we a ow proeed o show he mai resuls. As before, le us assume ha Ω 0 R is a ope se. We deoe is boudary by Γ 0 = Ω 0 ad irodue d = d Ω0 as he disae fuio of Ω 0. Reall also he defiiios of Ω ad Γ from 1). Lemma 4. For eah x R \ Ω 0, Furhermore, here exiss x 0 Γ 0 wih dx) = x x 0. dx) = if y Γ 0 x y. 9) Proof. See 2.2) o page 337 of [9] for 9). Γ 0 is losed, ad we a learly resri he ifimum o some bouded subse of Γ 0. Hee, his subse is ompa ad here exiss a miimiser x 0. I he followig, we are ieresed i esimaig he surfae area of Ω for > 0. Before we a do ha, le us briefly reall he appliable oeps for defiig suh a surfae area i he firs plae: For a ope se Ω R, we deoe by P Ω) is perimeer as defied, for isae, by Defiiio 3.35 o page 143 of [3]. Noe ha we are mosly ieresed i he perimeer relaive o he base se R.) The se Ω is said o have fiie perimeer or o be a Caioppoli se if P Ω) <. Furhermore, le us irodue also he Hausdorff measure followig Defiiio 2.46 o page 72 of [3]: 6

7 Defiiio 2. Le k N ad Ω R. For δ > 0, we defie { ) k } Hδ k di Ω) = if ω k 2 Ω U i, d i = sup x y, d i 2δ. x,y U i i=1 i=1 Here, ω k deoes he volume of he k-dimesioal ui ball as i Lemma 1. The value d i is he diameer of he se U i, ad i is allowed o be a mos 2δ i order for U i) o be a admissible δ-overig of Ω. Furhermore, he k-dimesioal Hausdorff measure of Ω is he give by H k Ω) = sup Hδ k Ω) = lim Hk δ>0 δ 0 + δ Ω). Noe ha we defie he Hausdorff measure i suh a way ha H orrespods o he -dimesioal Lebesgue measure. For a proof, see Theorem 2.53 i [3].) This is he reaso for iludig ω k i he defiiio. Oher auhors e. g., [16]) do o add his ormalisaio osa, whih resuls i a oio of H k ha is differe from Defiiio 2 by a osa. For he ase of oly oe dimesio, he siuaio is simple sie ses of fiie perimeer i oe dimesio a be represeed up o a se of measure zero) as he uio of a fiie umber of iervals: Theorem 2. Le = 1 ad Ω 0 R be ope ad bouded. The Γ is a fiie se for eah > 0 ad is ardialiy is o-ireasig wih respe o. Furhermore, H 0 Γ ) P Ω 0). 10) If Ω 0 has fiie perimeer ad is suffiiely small, he boh values are aually equal. Proof. Le > 0 ad x Γ. Lemma 4 implies ha here exiss x 0 Γ 0 wih x x 0 =. Assume, wihou loss of geeraliy, ha x 0 < x. I follows ha I x = x 0, x) d 1 0, )). Furhermore, if y Γ ad x y, he I x I y =. Sie vol I x) = > 0 for eah x Γ ad Ω is bouded, he ardialiy of Γ is bouded as H 0 Γ ) vol Ω ) / ad hus fiie. If we have 0 < s <, he esimae 10) implies ha H 0 Γ ) P Ω s) H 0 Γ s). Hee i follows ha he ardialiy is o-ireasig whe we have esablished 10). For 10), assume ha Ω 0 has fiie perimeer he siuaio is rivial oherwise). Aordig o Proposiio 3.52 o page 153 of [3], here exis p N ad p disjoi iervals J i = [a i, b i] suh ha Ω 0 p i=1 Ji. These wo ses a oly differ up o a se of measure zero. Furhermore, P Ω 0) = 2p. As before, we a assoiae a ierval I x d 1 0, )) o eah x Γ, ad all I x are disjoi. If we assume ha I x = x 0, x), he x 0 = b i for some 1 i p. Similarly, x 0 = a i if I x = x, x 0). This implies 10), sie If we assume a orderig suh as ad deoe by H 0 Γ ) 2p = P Ω 0). a 1 < b 1 < a 2 < b 2 < < a p < b p L = if ai+1 bi) > 0 i=1,...,p 1 he miimal disae bewee he iervals J i, he equaliy holds wih H 0 Γ ) = 2p for < L/ A Boud o he Hausdorff Measure Iuiively, Ω is osrued from Ω 0 by addig a layer of hikess oo Γ 0. Followig his piure, oe a imagie ha he volume of his layer should roughly equal imes he surfae area i. e., perimeer) of eiher Ω 0 or Ω. This argume a be made rigorous by esimaig he volume i erms of P Ω 0), ad he H 1 Γ ) i erms of he volume. The former will be doe i Subseio 4.2. We will show he laer as our firs mai resul i his subseio. This is, somehow, a iverse isoperimeri iequaliy. Of ourse, i he geeral siuaio o iverse o he lassial isoperimeri iequaliy see Subseio of [11]) holds. I our ase, however, i works beause he osidered volume is o allowed o be arbirarily hi. Defiiio 3. For a fixed iiial se Ω 0 ad > 0, we defie he ewly reaed volume o be ) U = B x 0) \ Ω 0 = {x R 0 < dx) < }. x 0 Γ 0 We a ow sae ad prove he firs mai resul: 7

8 Theorem 3. There exiss a dimesioal osa C suh ha holds for all > 0. P Ω ) H 1 Γ ) C vol U) Proof. The firs iequaliy is a well-kow fa abou he relaio bewee perimeer ad he Hausdorff measure. See, for isae, Proposiio 3.62 o page 159 of [3]. We will ow show he seod iequaliy. For his, le δ > 0 be give. The learly Γ x Γ B δ x). Aordig o Viali s overig heorem see Theorem 1 o page 27 of [11]), here exiss a ouable subse X Γ suh ha Γ B 5δ x) 11) x X ad all B δ x) are disjoi for x X. Noe ha X is, i fa, fiie if Ω 0 ad hus also Γ are bouded. For eah x Γ, here exiss a orrespodig x 0 Γ 0 wih x x 0 = aordig o Lemma 4. Furhermore, y y 0 for all y Γ ad y 0 Γ 0. For x X ad is assoiaed poi x 0 Γ 0, le us defie S x = S φδ) x 0, x). Noe ha he odiio 8) is saisfied for eah pair S x, S y) wih x, y X, so ha all S x ad S y wih x y are disjoi by Lemma 3. Also oe ha a basi geomeri argume implies S x Ω 0 = for small eough δ. Thus, we fid ha eah S x is oaied i he ewly reaed volume ad ge ) vol S x) = vol S x vol U ). 12) x X Sie he elarged balls i 11) provide a pariular 5δ-overig of Γ, we kow ha x X H 1 5δ Γ ) 5δ) 1 ω C vol S x). x X x X The las esimae ad he osa C ome from Lemma 2. Togeher wih 12), his yields H 1 5δ Γ ) 5 1 C vol U). The boud o he righ-had side does o deped o δ ay more, so ha we a ake he limi δ 0 + o fiish he proof. Havig his firs resul, we a already show ha all evolved ses Ω mus be Caioppoli ses: Corollary 1. Le Ω 0 be bouded. The Ω has fiie perimeer for all > 0. Proof. From he boudedess of Ω 0, we a direly olude ha also Ω ad U are bouded ses for ay fixed. Thus, vol U ) < ad Theorem 3 implies ha H 1 Γ ) is fiie for eah. I follows ow agai from Proposiio 3.62 o page 159 of [3] ha Ω is a se of fiie perimeer. Take oe ha he aual boud we ge from Corollary 1 diverges like 1/ for 0 +. I will be he fous of he ex subseio i pariular, Corollary 2) o show a uiform boud as 0 + uder addiioal assumpios. Wihou hese assumpios, however, we a o hope for ay srog improveme of Corollary 1: As we have see i Theorem 1, he opimal upper boud mus diverge sroger ha 1/ s for ay s 0, 1). 4.2 Uiform Bouds As we have see above i Theorem 3, he quaiy vol U ) / is ruial as i gives a upper boud o he evolved ses perimeers. Pariularly ieresig is he limi 0 +. As our seod mai resul below, we a show ha here exiss a uiform upper boud for 0 + as log as a uiform desiy odiio holds for he iiial se Ω 0. This odiio preves arbirarily sharp orers ad usps. To be preise: 8

9 Defiiio 4. Le A Γ 0, 0, 1) ad 0 > 0. We say ha Ω 0 has 0, )-uiform lower desiy o A if he esimae vol B x) Ω0) 0 < 13) vol B x)) holds for all 0, 0) ad x A. Similarly, Ω 0 is said o have 0, )-uiform upper desiy o A if vol B x) Ω 0) vol B x)) 1 < 1. 14) Whe boh odiios are saisfied ogeher, Ω 0 simply has 0, )-uiform desiy o A. For fixed x ad i he limi 0 +, he quoie i 13) ad 14) gives he desiy of Ω 0 a x. See page 158 of [3] for some kow resuls abou his quaiy. I pariular, le FΩ deoe he redued boudary of a ope se Ω. Roughly speakig, his is he se of all boudary pois where a measure-heorei varia of he ormal veor o he boudary a be defied. See Defiiio 3.54 o page 154 of [3].) The Ω has desiy 1/2 a all pois i FΩ. This is, for isae, also rue i he example osrued i Seio 2. Hee, oe ha uiformiy of he esimaes is really ruial for our purposes i he followig. Noe ha we are o he firs o irodue he oep of uiform lower desiy. I has bee used already by ohers i a similar oex. See, for isae, Proposiio 4.2 i [2] ad Theorem 6 i [7]. The relaio bewee uiform desiy ad oher, more esablished geomeri properies will be disussed i more deail i Subseio 4.3. For our esimae of vol U ), we eed o somehow ge a upper boud o i erms of he perimeer of Ω 0. For a lassial resul i his direio, see 3.54) o page 156 of [3]. Uforuaely, his esimae is loal i aure ad o uiform over he whole boudary of Ω 0. Noe, however, ha 13) ad 14) ogeher are equivale o mi vol B x) Ω0), vol B x) \ Ω0)). 15) vol B x)) This relaio a be ombied wih he relaive isoperimeri iequaliy see, for isae, Theorem 2 o page 190 of [11]) o ge he uiform esimae ha we eed: Lemma 5. Le Ω 0 have 0, )-uiform desiy o A. The here exiss a dimesioal osa C suh ha for all x A ad 0, 0). 1 C Proof. Sie we assume uiform desiy, 15) implies ha ) 1 1 H 1 B x) FΩ 0) vol B x)) = ω mi vol B x) Ω 0), vol B x) \ Ω 0)) for all 0, 0). If we also apply he relaive isoperimeri iequaliy, we ge 1 C ) 1 1 P Ω0; B x)) for some dimesioal osa C. This implies he resul ogeher wih he well-kow relaio bewee perimeer ad H 1 ha a be foud i Theorem 3.59 o page 157 of [3]. So far, we have assumed uiform desiy of Ω 0. I will ur ou, however, ha i is eough o require oly uiform lower desiy. Uiform upper desiy is provided auomaially if we hoose he subse A Γ 0 i he righ way: Defiiio 5. We say ha x 0 Γ 0 is bakwards reahable for ime > 0 if here exiss x R wih The se of all bakwards reahable pois for ime is deoed by R. x 0 x = dx). 16) See Figure 4a for a illusraio of he se R : The poi x 0 R is show ogeher wih a possible x Γ ha fulfils 16). Noe ha oly he red par of Γ 0 is bakwards reahable. Thus, we see ha R is aually more regular ha Γ 0 iself. I pariular, Ω 0 has always uiform upper desiy o R. To udersad why his mus be he ase, ake a look a Figure 4b: Wheever x 0 ad x are as idiaed, he ball B x) mus be disjoi o Ω 0 sie oherwise dx) < would be he ase. Thus, he volume of B x) B x 0) a ever be par of Ω 0, whih implies a upper boud for he desiy of Ω 0 a x 0. For he show siuaio, he desiy is aually 1/2. The maximal possible desiy would be ahieved if also he ligh grey area were par of Ω 0.) Le us formalise his argume: 9

10 x x Γ x 0 R x 0 Ω 0 a) The se R of Defiiio 5. b) The mai argume i he proof of Lemma 6. Figure 4: The bakwards reahable se ad is regulariy wih respe o uiform upper desiy. The dark grey regio is Ω 0. The poi x 0 is o R Γ 0, wih x Γ suh ha 16) holds. Lemma 6. Le > 0 ad R be he bakwards reahable se for ime. The Ω 0 has, )-uiform upper desiy o R, where is a dimesioal osa. Proof. Le e R be arbirary wih e = 1. We defie 0 < = vol B1 0) B1 e)) vol B 1 0)) Now hoose x 0 R ad τ. We have o show ha 14) holds for B τ x 0) wih he defied. By Defiiio 5, here exiss x R suh ha τ x 0 x = dx). We a assume, wihou loss of geeraliy, ha x 0 x = τ. Cosiderig Figure 4b, his implies B τ x) Ω 0 =. Hee: vol B τ x 0) Ω 0) vol B τ x 0)) = 1 vol Bτ x0) \ Ω0) vol B τ x 0)) 1 < 1. vol Bτ x0) Bτ x)) vol B τ x 0)) = 1 Aoher impora observaio is ha he bakwards reahable se is already suffiie for he osruio of he ewly reaed volume U. This allows us o resri our osideraios o he more regular R isead of Γ 0 iself laer o. Lemma 7. For 0 < s <, R R s. Furhermore, U \ U s x 0 R s B x 0). Proof. The ilusio R R s is immediaely lear from Defiiio 5. Pik x U \ U s arbirarily. By Lemma 4 we a fid x 0 Γ 0 wih dx) = x 0 x. Moreover, x U s implies ha dx) s, so ha x 0 R s. Similarly, x U yields dx) < ad hus x B x 0). Wih his resul, all preparaios are i plae ad we a proeed o he aual esimae of vol U ). This is doe i wo seps: Firs, we esimae vol U 2 \ U ). The regulariy of he bakwards reahable se wih respe o uiform upper desiy of Ω 0 a be used for his siuaio. Aferwards, we build he uio of a sequee of suh srips i order o ge vol U ) iself. Lemma 8. Assume ha Ω 0 has 0, )-uiform lower desiy o Γ 0. The here exiss a dimesioal osa C suh ha vol U 2 \ U ) C ) 1 P Ω0) holds for all 0, 0). 10

11 Proof. Aordig o Lemma 6, we kow ha Ω 0 has, )-uiform upper desiy o R wih some dimesioal. Sie i has uiform lower desiy per assumpio, i has, )-uiform desiy boh upper ad lower) wih = mi, ). Furhermore, oe ha 1 = 1 mi, ) Thus Lemma 5 implies ha C ) 1 H 1 B x 0) FΩ 0) for all x 0 R wih some dimesioal C. Takig i eve furher, his yields also ) ) vol B 10 x 0) = 10 vol B x 0) C ) 1 H 1 B x 0) FΩ 0) 17) for ye aoher dimesioal osa C. Makig use of Lemma 7, we kow ha U 2 \ U x 0 R B 2 x 0). Wih Viali s overig heorem see, agai, Theorem 1 o page 27 of [11]), we a osru X R a mos ouable suh ha he ses B 2 x 0) are disjoi for x 0 X, bu sill U 2 \ U B 10 x 0). x 0 X Takig he measure o boh sides of his ilusio ad usig 17), we fially fid vol U 2 \ U ) C x 0 X ) vol B 10 x 0) C ) 1 H 1 B x 0) FΩ 0) x 0 X ) 1 H 1 FΩ 0) = C The simplifiaio of he sum is jusified beause all ses B x 0) are disjoi. Theorem 4. Le Ω 0 have 0, )-uiform lower desiy o Γ 0. The vol U ) for all 0, 0) ad a dimesioal osa C. C ) 1 P Ω0) Proof. Le 0, 0) be give. The he disjoi elesopi deomposiio holds. Togeher wih Lemma 8 his yields vol U ) = k=1 This fiishes he proof. ) 1 P Ω0). U = ) ) U \ U /2 U/2 \ U /4 = U2/2 k \ U /2 k) vol ) U 2/2 k \ U /2 k C ) 1 P Ω0) k=1 k=1 2 = C ) 1 P Ω0). k Whe we ombie Theorem 4 wih Theorem 3, we fially ge a uiform boud for H 1 Γ ). This resul is very similar o Theorem 6 i [7], bu oe ha i holds for all 0 ad o jus for almos all: Corollary 2. Assume ha Ω 0 has 0, )-uiform lower desiy o Γ 0 ad ha Ω 0 is bouded. I pariular, le Ω 0 B R 0) for some R > 0. The H 1 Γ ) C 1 + P Ω 0) + 1) 18) for all 0. The osa C depeds oly o, 0, ad R bu o oher properies of Ω 0. 11

12 Proof. Noe ha he siuaio is lear for = 0 as log as we hoose C 1. From Theorem 4, we kow ha vol U ) / C P Ω 0) for all 0, 0). Furhermore, sie Ω 0 B R 0), oe ha Ω B R+ 0). Thus, for 0, ) vol U ) ω R + ) C 1 + vol U) ) C C 1 + 1). The laim ow follows from Theorem 3, if we ombie boh esimaes for vol U ) /. 4.3 Geomeri Regulariy Properies i he Lieraure The mai igredie for he resuls i he previous Subseio 4.2 is a pariular geomeri propery of he iiial se Ω 0, amely uiform desiy from Defiiio 4. As poied ou above, his oio has bee used already by ohers o ahieve similar resuls see [2] ad [7]). We are o aware of ay appliaios i a broader oex, hough. Thus, i makes sese o pu i io perspeive wih similar geomeri properies ha are more esablished i he lieraure ad more widely used. I pariular, a variey of so-alled uiform) segme ad oe properies is ofe used o haraerise geomeri regulariy of ses. For a horough iroduio, see Seio 2.6 of [9]. Sie uiformiy plays a impora role for he resuls of Subseio 4.2, i makes oly sese o osider he uiform varias of hose segme properies. All o-uiform properies are fulfilled by he example developed i Seio 2, sie i is osrued oly from irles.) Furhermore, he uiform fa) segme propery aloe also provides very lile regulariy. For isae, a usp saisfies i while i learly does o have uiform lower desiy. Thus, le us fous o he uiform oe propery. For oveiee, we reall Defiiio 6.3 o page 115 of [9]: Defiiio 6. For x 0, x R ad φ [0, π/2], defie he ope oe C φ x 0, x) = { y R x 0 y x 0 x os φ < y x 0) x x 0) < x 0 x 2 }. This is similar o he seor S φ x 0, x) of Defiiio 1 sudied above, bu i desribes a oe wih fla base, i. e., wihou a spherial ap. Now, le Ω R be ope. We say ha Ω saisfies he uiform oe propery if here exis > 0, φ 0, π/2) ad ρ > 0 suh ha for all x 0 Ω here is x R wih x 0 x = ad for all d B ρ 0). x + d Ω C φ x 0 + d, x + d) Ω Sie he uiform oe propery esures for eah boudary poi he exisee of a oe ha is eirely oaied i Ω, we a use his oe s volume as a lower boud o he desiy of Ω. Thus, he uiform oe propery is a sroger odiio ha uiform lower desiy: Theorem 5. Le Ω R saisfy he uiform oe propery wih ad φ as i Defiiio 6. The Ω has, rφ))- uiform lower desiy o Ω. Similarly, if R \ Ω has he uiform oe propery wih hese osas, he Ω has, rφ))-uiform upper desiy. Proof. Le x 0 Ω be give. Aordig o Defiiio 6, here exiss x R wih x 0 x = suh ha C φ x 0, x) Ω. Noe ha S φ x 0, x) C φ x 0, x) sie Thus, for eah τ 0, ), learly Hee, we a esimae x 0 y < x 0 x y x 0) x x 0) y x 0 x x 0 < x 0 x 2. B τ x 0) S φ x 0, x) B τ x 0) Ω. vol B τ x 0) Ω) vol B τ x 0) S φ x 0, x)) = rφ) vol B τ x 0)) based o 6). This shows he laim. The proof for uiform upper desiy works aalogously. Aoher oep relaed o our defiiio of uiform lower desiy are ses wih fiie desiy perimeer as defied i [4] ad Subseio 3.1 of [8]: Defiiio 7. Le Ω R be ope ad h > 0. The h-desiy perimeer of Ω is he defied as where V ɛ Ω) is he ɛ-evelope of Ω: V ɛ Ω) = P h Ω) = x Ω sup 0<ɛ<h If P h Ω) is fiie, we all Ω a se of fiie h-desiy perimeer. vol V ɛ Ω)), 19) 2ɛ B ɛ x) = {x R d Ω x) < ɛ} 12

13 This a be ierpreed as a relaxaio of he 1)-dimesioal Mikowski oe see, for isae, i [12]). To be preise, he Mikowski oe resuls if he supremum i 19) is replaed by he limi ɛ 0 +. I is easy o see ha V ɛω 0) is relaed o he ewly reaed volume U ɛ of Defiiio 3: The se U ɛ is he par of V ɛω 0) whih is ouside of Ω 0. Hee, a argume similar o he proof of Theorem 4 a be applied o show ha uiform desiy implies fiie desiy perimeer. Akowledgemes The auhor would like o hak Wolfgag Rig of he Uiversiy of Graz for horough proofreadig of he mausrip. This work is suppored by he Ausria Siee Fud FWF) ad he Ieraioal Researh Traiig Group IGDK Referees [1] Giovai Alberi, Sefao Biahii, ad Gialua Crippa. Sruure of Level Ses ad Sard-Type Properies of Lipshiz Maps. Aali della Suola Normale Superiore di Pisa - Classe di Sieze, 124): , [2] Fred Almgre, Jea E. Taylor, ad Lihe Wag. Curvaure-Drive Flows: A Variaioal Approah. SIAM Joural o Corol ad Opimizaio, 312): , Marh [3] Luigi Ambrosio, Niola Fuso, ad Diego Pallara. Fuios of Bouded Variaio ad Free Disoiuiy Problems. Oxford Mahemaial Moographs. Oxford Siee Publiaios, [4] Dori Buur ad Jea-Paul Zolésio. Free Boudary Problems ad Desiy Perimeer. Joural of Differeial Equaios, 126: , [5] Mari Burger. A Framework for he Cosruio of Level Se Mehods for Shape Opimizaio ad Reosruio. Ierfaes ad Free Boudaries, 5: , [6] Mari Burger, Norayr Maevosya, ad Marie-Therese Wolfram. A Level Se Based Shape Opimizaio Mehod for a Ellipi Obsale Problem. Mahemaial Models ad Mehods i Applied Siees, 214): , April [7] David G. Caraballo. Areas of Level Ses of Disae Fuios Idued by Asymmeri Norms. Paifi Joural of Mahemais, 2181), Jauary [8] Mihel C. Delfour ad Jea-Paul Zolésio. The New Family of Craked Ses ad he Image Segmeaio Problem Revisied. Commuiaios i Iformaio ad Sysems, 41):29 52, [9] Mihel C. Delfour ad Jea-Paul Zolésio. Shapes ad Geomeries: Meris, Aalysis, Differeial Calulus, ad Opimizaio. Advaes i Desig ad Corol. SIAM, seod ediio, [10] Mar Droske ad Wolfgag Rig. A Mumford-Shah Level-Se Approah For Geomeri Image Regisraio. SIAM Joural o Applied Mahemais, 666): , [11] Lawree C. Evas ad Roald F. Gariepy. Measure Theory ad Fie Properies of Fuios. Sudies i Advaed Mahemais. CRC Press, [12] Herber Federer. Geomeri Measure Theory. Spriger, [13] Daiel Kraf. A Hopf-Lax Formula for he Time Evoluio of he Level-Se Equaio ad a New Approah o Shape Sesiiviy Aalysis. Prepri IGDK , hps://igdk1754.ma.um.de/foswiki/ pub/igdk1754/prepris/kraf_2015a.pdf. [14] Daiel Kraf. A Hopf-Lax Formula for he Level-Se Equaio ad Appliaios o PDE-Cosraied Shape Opimisaio. I Proeedigs of he 19h Ieraioal Coferee o Mehods ad Models i Auomaio ad Robois, pages IEEE Xplore, [15] Saley J. Osher ad James A. Sehia. Fros Propagaig wih Curvaure-Depede Speed: Algorihms Based o Hamilo-Jaobi Formulaios. Joural of Compuaioal Physis, 79:12 49, [16] James Yeh. Real Aalysis: Theory of Measure ad Iegraio. World Sieifi, seod ediio,