Continuity of the visibility function

Publicacions Ma emá iques, Vol 35 (1991), 323-332 . CONTINUITY OF THE VISIBILITY FUNCTION Abs ac ANA FORTE CUNTO G . Bee de ined he isibili y unc ion o a se S and p o ed i s con inui y in he in e io o S . I is p o ed he e ha he isibili y unc ion o a plana Jo dan domain S is con inuous p ecisely a he cone poin s o he bounda y o S . 1 . No a ions and basic de ini ions Unless o he wise s a ed, all he poin s conside ed he e a e included in he Euclidean plane E2 . The complemen , in e io , closu e, bounda y and con ex hull o a se S a e deno ed by C S, in S, cl S, bd S and con S, espec i ely . The open segmen joining x and y is deno ed (x y) . The subs i u ion o one o bo h pa en heses by squa e ones indica es he adjunc ion o he co esponding endpoin s . The ay issuing om x and going h ough y is deno ed R(x --> y), while R(yx -) is he ay issuing om x and going in he opposi e di ec ion o ha o R(x --~ y) . Rays a e always closed . We say ha x sees y ia S i [x y] C S . The s a o x in S is he se s (x, S) o all he poin s o S ha see x ia S . A s a -cen e o S is a poin x E S such ha s (x, S) = S . The con ex ke nel o S is he se ke S o all he s a -cen e s o S . S is s a shaped i ke S ~ 0 . A Jo dan domain is a compac connec ed se o E2 whose bounda y is homeomo phic o he uni ci cle . The open and closed disks o cen e x and adius b will be deno ed U(x ; S) and B(x ; 6), espec i ely . I yE bd S and x E s (y, S) we say ha he ay R(x -> y) is inwa d h ough y i he e exis s E R(xy -~) such ha (y ) C in S . O he wise we say ha R(x , y) is ou wa d h ough y . The inne s em o y wi h espec o S is he se ins(y, S) o med byyand all he poin s o s (y, S) ha issue ou wa d ays h ough y . A poin x E S is a poin o local con exi y i he e ex is s E > 0 such ha S n B (x, E) is con ex . O he wise, x is a poin o local noncon exi y . We ema k ha he dis inc ion is signi ican only o bounda y poin s, since e e y in e io poin is i ially o local con exi y . The se o all poin s o local con exi y o S and ha o all poin s o local noncon exi y a e deno ed le S and lnc S, espec i ely . I is easy o see ha le S is open and lnc S is closed in he ela i e opology o bd S . An obs uc ion xone is l a connec ed componen o lnc S . 32 4  A . FORTE CUNTO A poin x E bd S is a la poin i x E le S n le CS . The se o all such poin s is deno ed ip S . x E bd S is an in lec ion poin (and he se o all in lec ion poin s is deno ed i p S) i eí he x E lnc S n lnc CS o x E lnc S n le CS n cl( lp S) . An a e I' C bd S keeps he sense o cu a u e i ei he I' C le S o I' C lnc S . I S is a closed se wi h nonemp y in e io and x E S, hen he se o c i ical isibili y o x in S is he se c (x, S) = in S n bd s (x, S) . Each poin o his se is a poin o c i ical isibili y o x in S .  The poin x E S is clea ly isible om y ia S i he e exis s e > 0 such ha B (x, E) n S C s (y, S) . 2 . S a emen o he p oblem In [1] G . Bee de ines he isibili y unc ion as he one ha assigns o each poin x o a ixed measu able se S in he Euclidean space E ., he Lebesgue ou e measu e o s (x, S) . We shall deno e i (x) . In [1], [2]' and [3] se e al heo ems abou he con inui y o (x) in open se s, o in he in e io o he se s conside ed, a e demons a ed . The pu pose o he p esen pape is o s udy he beha io o he isibili y unc ion in he bounda y o a Jo dan domain S . The s udy is es ic ed o his case in o de o a oid di icul ies as hose p esen ed in he examples o [2] and [3] . In his case, he heo ems shown in [1] assu e he con inui y o (x) in in S . Fu he mo e, he bounda y cu e mus ha e ini ely many in lec ion poin s in he smoo h case, and ini ely many angula poin s in he nonsmoo h case . This will p e en he exis en e o singula poin s (Le . poin s o accumula ion o in lec ion poin s o angula poin s) . The s udy o he s a o a singula poin seems almos unmanageable o his au ho . A Jo dan domain wi hou singula poin s will be a egula Jo dan domain . We make a local s udy o he s a 's measu e in a poin x E bd S using he domain o good beha io N, ha is a neighbo hood o x ha ing he ollowing cha ac e is ics : i) i s cen e will be x, ii) N includes nei he in lec ion poin s no angula poin s o bd S excep possibly x i sel . Clea ly ii) assu es ha each o he wo suba cs I'1 and I'2 o N n bd S, ha ing x as one endpoin , keeps he sense o cu a u e . We gene alize his local esul s using he ac ha he s a s a e ans spanned by S . The de ini ion o his concep is gi en below . Lemma 3 .1 . Le S be a closed se o he plane, {x ; y ; z} C S such ha [x y] U [y z] CS . Le T= con {x ; y ; z} ha e a mos one poin w E lnc S such ha w E (x z) . Then T C S . P oo : A sligh a ia ion in he p oo o Co olla y 2 o [6] yields his lemma ha is, in he same spi i o Valen ine's esul , an use ul consequence o Tie ze's heo em on local con exi y . Lemma 3 .2 . Le S be a egula Jo dan domain and xo E bd S . The e exis s a domain o good beha io N = B(xo, S) . Fu he mo e, A= [B(xo, b)- {xo}] n bd S consis s o wo connec ed a cs ending a xo, such ha each o hem keeps he sense o cu a u e . P oo . De ine CONTINUITY OF THE VISIBILITY FUNCTION  325 3 . Auxilia y esul s K = {z E bdSIz is an in lec ion poin o an angula poin } 5 = d(x o , K - {xo}) Rom he inexis en e o singula poin s i ollows ha 5 > 0 . Le F be he connec ed componen o [bd S 1 B(xo, S/2)] ha includes xo . Then F = F 1 U 2 whe e each o hese suba cs ends a xo and keeps he sense o cu a u e . De ine B, = B(xo,5/2n) and I'1,, F2n as he connec ed componen s o F1 n B, and F2 n B espec i ely, ha include xo . Le A,=F n - [F1 nU F2 n ] . Owing o he simplici y o bd S, he e exis s a posi i e in ege m such ha B  , 10  , = 0 . The ball N = B  ,, sa is ies he hesis . In he sequel, he domain o good beha io wi h espec o xo will be deno ed by N . Le xo E bd S and L a be a line h ough xo . The maximal segmen de e mined by L,, in S is he connec ed componen la o L . 1 S ha includes xo . The union o all hose maximal segmen s is he an in xo spanned by S . The angula ampli ude o a an S(aaS) will be he no malizad Lebesgue measu e (in bd N o he adial p ojec ion o S om xo o a bd N . S will be an angula connec ed an i ha p ojec ion is connec ed in he ela i a opology o bd N . Lemma 3 .3 . ins(xo, S) and s (xo, S) a e ans in xo . P oo . Bo h se s a e s a shaped and xo is a s a -cen e o each o hem . Lemma 3 .4 . I = N n ins(xo, S) is an angula connec ed an . P oo . We conside wo al e na i as : (i) Le u E I, E I and xo be no collinea wi h hese poin s . The e exis u' E R(uxo ->) n N and ' E R( xo -) 1 N such ha (xo u'] n in S = 0 32 6  A . FORTE CUNTO and (xo '] n in S=~ . Assume ha (xo u') and (xo ') a e bo h included in N nC S . De ine /0 = min {d(xo, u) ; d(xo, ) ; d(xp, u') ; d(xp, ') } B' = B(xo, P/2)  ,  Bi = bd B', ui E [xp u] n Bi  ,  i E [xo ] n Bi, u1 E [xo u'] n B'  ,  1 E [xo '] n Bi . Clea ly (xo ui] C C S, and (xo i] C CS . Fú he mo e, condi ion (i) implies ha i 1 L(xo ui), hence [ i ui] n bdS =,A 0 . An analogous a gumen shows ha [ui i] n bd S 0 . Le qi E [ i ui] n bd S and pi E [ui i] n bd S . Hence pi =A xo and qi ~ xp by condi ion (i), and bd S canno c oss nei he (xo ui) no (xo i) since hese segmen s a e en i ely included in CS . A simila a gumen shows ha bd S canno c oss nei he (xo ui) no (xo i) . Since N is he domain o good beha iou o xp, only wo suba cs o bd S (call hem i and 2), bo h ha ing xo as one ex eme, a e included in N . De ine wo ci cula sec o s o B' : such ha qi E Si and pi E S2 . I ollows ha (1)  i C Si, I'2 C S2 and (2) (3) Si = ( i xo ui) and S 2 = (ui xo i) in (con {ui ; i ; xo}) n bd S= 0 in (con {ui ; i ; xo}) n bd5 = 0 I z E [ui i], i ollows om (2) and Lemma 3 .1 ha [z xo] C S and z E s (xp, S) .  De ine now z' E (ui i) n R(zxo ->) .  Using (3) and Lemma 3 .1 we ob ain ha (xo z') C C S, and z E ins(xo, S) .  A e y simila a gumen h,)lds when one o bo h o he segmen s [xo ui], [xo i] is included in bdS . We co-lclude ha I is con ex, whence i s adial p ojec ion om xo on o bd N mus be connec ed in he ela i e opology o bd N . (ii) Le u E I, E I, and xo E (u ) . Since R(uxo ->) is ou wa d, xo E (u ) and [xo ] C S, i ollows ha [xo ] C bd S . The same a gumen p o es ha [xo u] C bd S, and as S is a Jo dan domain, I n B' mus be a hal -ci cle . Clea ly, i s p ojec ion om xo on o bd N mus be connec ed . Lemma 3 .5 . J = N n s (xo, S) is an angula connec ed an . P oo . We conside h ee al e na i es : (1) Le u E J, E J and u E N n s ( , S), while xo ~ [u ] .  Le A_ con ({u ; ; xo}) .  Then in A n bd S = 0 since any c ossing o bd S o e CONTINUITY OF THE VISIBILITY FUNCTION  32 7 [xo u], . [xo, ] o [u ] would  uin he condi iona o isibili .- I z E [u u] ;, . by Lemma 3 .1 ollows ha z E J . Hence, J esul s con exa and i s adial p ojec ion om xo o e bdN mus be connec ed . (2) Le u E J, E J, u 0 N n s ( , S), while xo 1 [u ] . De ine b = min {d(u, x o ) ; d( , xo) } , B' = B(x o , 8/2), Bi = bd B', Clea ly u' E J, ' E J, and we mayasume ha u' 1 N n s ( ', S), since o he wise we would be in he si ua ion o pa (1) . Le z E (u ') l C S, p E ( ' z) l bd S l N l s ( ', S), u E [xo u] l Bi, ' E [xo ] l Bi . gE(u'z)nbdS(1N ls (ú,S) . The isibili y condi ions assu e ha p =~ xo, q =~ xo and p =,,= q . We in end o p o e ha p and q belong o di e en bounda y a es sepa a ed by xo in bd Sn N . I may happens ha p = ' o q = u', bu in his case one o he bounda y a es would be a segmen , and since xo 1 [p q], hese poin s mus belong o di e en a es . Le us assume ha p :pÉ ' and q :~ u', and suppose ha p and q belong o he same bounda y a e F l . Wi hou loss o gene ali y assume 6, ha q E a e (xo, p) C 1 . By he de ini ion o p, i is no he las poin o 1 in B' . Le A= con ({ ' ; p ; xo}) . We obse e he posi ion o I'1 wi h espec o A . I Pl l in A= ¢, ei he a double poin o a o bidden change in he sense o cu a u e would appea in bd S n N . I 1 n in A :~ 0, i would imply a 32 8  A . FORTE CUNTO double poin , a con adic ion o he isibili y condi ions o a o bidden changa o cu a u a . Hence, p and q mus belong o di e en bounda y a cs . Le Sl = ( ' xo u') he ci cula sec o o B' ha includes z ; S2 = B' n CS 1 . Then bd S n B' C S i . De ine z' ER(zxo -~) n Bi . Then z' E J, and z' E in S2 . Using pa í (1) we ob ain ha [u' z'] U [z' '] C J, and he adial p ojec ion o his union o e bd N is connec ed . (3)Le uEJ,VEJbu xoE[u ] . Le B', Bi, u' and ' be as in (2) . Le z E CS n B', and ake p E [u z] n s (u', S) n bd S q E [ ' z] n s ( ', S) n bd S Since z, u and a e no collinea , i ollows ha p 7~ q . Using ha z' _ R(zxo -) n Bi, and he same a gumen s as in (2) we ob ain [u' z'] U [z' '] C J . Lemma 3 .6 . Le {x n I n E NI } be a sequence o poin s in S such ha lim x  , _ xo . Then ins(xo, S) C  u n s (xj, S)  U {xo} U Q= ~~ lim  s (x, S) J U {xo} U Q, =1j=  n ', whe e Q is included in he union o a ani e numbe o maximal segmen s wi h espec o xo and has null measu e . P oo . Le pE ins(xo, S), p  xo . We conside wo al e na i as : a) I xo is clea ly isible om p, he e exis s a neighbo hood U(xo) such ha u(xo) n S C s (p, S) . Then i x,, -> xo, x,,, E u(xo) n Sdn > no and x n E s (p, S) b'n > no . Hence 00  00 00 pE  n s (x j , S ) C un s (xj, s ) . j=no . .  n=1 j=n b) I xo is no clea ly isible om p, i ollows om Theo em 2 .1 o [5] ha (p xo) n lnc S :,A 0 . Le z E (p xo) n lnc S . I z is a smoo h poin o bd S, he e exis s an obs uc ion zone I C lnc S ha includes z . Then con I' is suppo ed by, a mos , wo ays issuing om xo . Since he e a e ini ely many obs uc ion zones, and o cach o hem he e a e a mos wo maximal segmen s o c i ical isibili y, he amily o such segmen s is ini a . I z is no á smoo h poin o bd S, a simila a gumen , based in he de ini ion o egula Jo dan domain, assu es ha he amily o segmen s o c i ical isibili y is ini a . CONTINUITY OF THE VISIBILITY FUNCTION  32 9 Lemma 3 .7 . Le F be an angula connec ed an in xo and m(F) i s plana Lebesgue measu e . Then m(F) > 0 i and only i aa(F) > 0 . P oo : Taking xo as he o igin, he a ea o F can be easily compu ed by a posi i e adial unc ion (0) ha depends on he a gumen 0, whose ange o a ia ion is he p ojec ion o F om xo o e bd N . Le m and n be he endpoin s o his p ojec ion . I m, n, and xo a e no collinea , hen le A = con ({m ; xo ; n}), and a be he angle o med by R(xo -> n) and R(xo -> m) ex e io o A . Then, i is clea ha aa(F) = 27 -a . Hence, 27 (B) in (F) _  l  d dB « o Bo h implica ions o he hesis ollow eadily . O he wise, i m, n and xo a e in he same line L, ake E F- L, and A= con ({m, xo, n, }) . A e y simila a gumen holds and (1) is alid wi h a= 7 . Lemma 3 .8 . Le M and L be wo connec ed ans wi h e ex xo, M CL . Then m(M) < m(L) i and only i aa(M) < aa(L) . P oo . The se F= (L - M) U {xo} is a an in xo whose p ojec ion om xo o e bd N is no necessa ily connec ed, bu has a mos wo connec ed componen s . Hence F= Fl UF2 and L = M U Fl U F2 . Bo h implica ions o he hesis can be ob ained om equali y (1) o Lemma 3 .7 and he addi i i y o Lebesgue measu e . Lemma 3 .9 . m(ins(x, S) l N) < m(s (x, S)n N) implies ha m(ins(x, S)) < M(s (x, S)) . P oo . . Le us assume he exis en e o a sys em o pola coo dina es cen e ed a x and simila o he one desc ibed in Lemma 3 .7 . I a and /l a e he angula coo dina es o he endpoin s o he adial p ojec ion o ins(x, S) o e bd N, and a', /« a e he co esponding coo dina es o s (x, S), hen aa(ins(x, S)) = /p - a, aa(s (x, S)) = /3' - a' . Le e be he adius o N . F om Lemma 3 .8 i ollows ha a' < a < 3 < /j' . Using he no a ion o Lemma 3 .7 i esul s ha and a (0) m(s (x, S)) = m(s (x, S) 1 N) +  d dB «~ E 0 (0) m(ins(x, S)) = m(ins(x, S) n N) + 1.  d dB « E and he s ic inequali y o he hesis ollows eadily . 33 0 n-1 oo n-oo F om Lemma 3 .6 i ollows ha A . FORTE CUNTO 4 . The main heo em and i s co olla ies Theo em 4 .1 . Le S be a egula Jo dan domain and xo E bd S . Then, he ollowing s a emen s a e equi alen : (i) m(s (xo, S)) = m(ins(xo, S)) (ii) is con inuous in xo . P oo . (i) => (ii) . Assume ha is discon inuous in xo . Owing o he uppe semicon inui y o (Bee , [1], he e mus exis a sequence {xn} in S such ha xn -+ xo bu lim (x n ) < (xo) . Hence, lim m(S (x n , S)) < m(s (xo, S)) 00 00 ins(xo, S) C  U n s (xj, S)  U {xo} U Q n-1j-n whe e Q has null measu e . F om (2) i ollows ha 00 00 m(ins(xo, S)) < m  U I  I s (xj, S)  + m({xo}) + m(Q) _ n=1j=n 00 hmoo m I I  I s (xj, S) I <  hmoo M(S (x n , S)) . n  n- i=n F om his inequali y and (1) we ob ain a con adic ion o (i) . (ii) => (i) .  Assume ha m(s (xo, S)) =~ m(ins(xo, S)) .  I is clea ha m(s (xo, S)) - m(ins(xo, S)) > 0, and ha D = [s (xo, S) - ins(xo, S)] is a an a xo ha ing posi i e measu e . F om Lemma 3 .8 i ollows ha aa(s (xo, S)) > m(ins(xo, S)), whe eas om Lemmms 3 .3 and 3 .4 weknow ha bo h Nn ins(xo, S) and Nn s (xo, S) a e angula ly connec ed . Then, he p ojec- ion o D om xo o e Ni has a mos wo connec ed componen s, and a leas one o hem wi h posi i e measu e . Hence, i is clea ha m(s (xo, S) n N) > m(ins(xo, S) n N), and ha he e exis s a se A C D n N such ha in A =,4 0 . Selec E in A and w E ins(xo, S), and call L(w, xo) he line h ough w and xo and H+, H - he wo open semiplanes in which his line di ides E2 . Since V L(w, xo) we can assume ha E H+ .  The e exis s e > 0 such ha B( , E) CAC D n N .  Since R( - xo) is inwa d, he e exis a poin ' E R( xo ->) n H - n S n N wi h ( 'xo) C in S . Clea ly ' 1 ins(xo, S) and ha con adic s (ii) . CONTINUITY OF THE VISIBILITY FUNCTION  33 1 is no Cea ly isible o m ' . Le ,, =- ! ' + n- I1 xo . Hence dn n , E ( ' xo) and n --> xo . Fa he mo e, each o he n has he same isibili y es ic ions as ' wi h espec o . Le L( , xo) be he line h ough and xo, and U be he connec ed subse o S limi ed by L( , xo) and no isible om ' ia S . Fu he mo e, he poin s o U a e no isible ia S om each o he , Call S = S- U and le be he isibili y unc ion o S . Clea ly i holds dn ( n) = ( n), (xo) G V(X0) and om he uppe semicon inui y o we ob ain lim ( n) = lim ( n) < D(X0) G V(X0) n-Oo n-oo We say ha x E bd S is a cone poin i he e exis s a line L h ough x such ha s (x, S) is included in one o he wo closed semiplanes de e mined by L . Lemma 4 .2 . I S is a Jo dan domain and x is a bounda y poin o S, he e exis s a line L h ough x ha lea es ins(x, S) a one side o i . P oo .. As Lmma 3 .4 s a es, ins(x, S) n N is an angula connec ed an . I such a line does no exis , he e mus be a line h ough x ha in e sec s he adial p ojec ion om xo o ins(x, S) on bd N in wo poin s { , ; 2}, whe e a leas one o hem (say i) is no an endpoin o ha p ojec ion . Hence he e should exis o E (xo l) such ha (xo o) be included in o in S . Bu his should con adic he ac ha 2 E ins(x, S) . Theo em 4 .3 . The poin s o con inui y o he isibili y unc ion o S on he bounda y o S a e p ecisely he cone poin s o ha bounda y . P oo . . Le x be a cone poin o bd S, and L a line h ough x ha di ides E2 in o wo open semiplanes H+ and H - such ha s (x, S) C cl(H+) . Take u E s (x, S) - L, and le u' E R(ux --~) such ha (x u'] C H - . Since u' ~ s (x, S), we ha e wo al e na i es : (a) (x u') nS = 0 . (b) (x u') n S :,I= 0 and (x u') nCS  ~ . Each o his al e na i es p oduces easily a poin E (x u'] such ha (x ) n in S = 0, whence u E ins(x, S) . We ha e shown ha s (x, S) - ins(x, S) C L nS and he las se has null measu e . Hence x sa is ies hypo hesis (i) o Theo em 4 .1 and is con inuous a x . Con e sely, assume ha is con inuous a x . F om Lemma 4 .2 he e exis s a line L ha p oduces a closed semiplane H+ including ins(x, S) . Repea ing a gumen s used in he second pa o Theo em 4 .1 we ob ain ha he se D = s (x, S) - ins(x, S) is a an ha ing one o wo connec ed componen s and has null measu e . I ollows ha D C L, whence s (x, S) C H+ and x is a cone poin .