urvature and sopermetrc nequalty Julà ufí, Agustí Reventós, arlos J Rodríguez Abstract We prove an nequalty nvolvng the length of a plane curve and the ntegral of ts radus of curvature, that has as a consequence the - dmensonal Ros nequalty We also study the correspondng nequalty for the case the curve s a polygon For ths we must ntroduce a new concept of radus of curvature at the vertces of a polygon Fnally we provde the ln between the dscrete and the contnuous settngs Introducton The startng pont of ths note s the followng nequalty: f = K s the boundary of a compact, convex set K of area A n R, then A ρ(s) ds, () where ρ = ρ(s) < s the radus of curvature on and ds sgnfes arclength measure on Equalty holds f and only f s a crcle A proof of () s gven n [] On the other hand one has the sopermetrc nequalty Prepublcacó Núm 8, mag Departament de Matemàtques http://wwwuabcat/matematques A L 4π, where L s the length of, wth equalty f and only f s a crcle So t s natural to try to compare the two quanttes L /4π and ρ(s) ds In ths paper we prove that L 4π ρ(s)ds, wth equalty f and only f s a crcle (Theorem ) Next we study ths nequalty for the case the curve s a polygon For ths we need to ntroduce a noton of radus of curvature at the vertces of a polygon (Defnton ) Ths radus of curvature s a good approxmaton Partally supported by FEDER/Mcnn MTM 8-556-- Partally supported by FEDER/Mcnn MTM9-759
of the radus of curvature of a smooth curve (Theorem ) We prove the followng result: L 4π ρ + ρ + l, = where L s the length of the polygon, l the length of ts sdes, and ρ the curvature at ts vertces Equalty holds f and only f the polygon s umblcal (Theorem 3) As a consequence we obtan a dscrete proof of Theorem (Theorem 4) an a geometrcal proof of an nterestng analytcal property stated n Lemma Inequalty () s the two-dmensonal analogue of Ros nequalty: V 3 H da, where H > s the mean curvature of the boundary S of a compact doman of volume V n R 3, and da sgnfes area measure on S Equalty holds f and only f S s a standard sphere (see [] and [3]) On the ntegral of the radus of curvature We begn wth the followng analytcal result, suggested by the geometrcal problems we study here Lemma Let p : R R be a perodc functon of class wth perod π Then ( π p(φ) dφ) π S π (p(φ) + p (φ)) dφ Equalty holds f and only f p(φ) + p (φ) s constant Proof Note frst that π = π π [ π (p + p ) dφ ( p dφ + π π p dφ) (p ) dφ π ] (p ) dφ ( By developng p = p(φ) n Fourer seres expanson, p(φ) = a + (a n cos nφ + b n sn nφ), n= we have, by Parseval s dentty, π p dφ = a + (a n + b n), π and smlar expressons for p and p n= π p dφ) ()
Substtutng n () we have π n= π π (p + p ) dφ ( p dφ) [ ] = 4π a + (a n+b n)+ n 4 (a n + b n) n (a n+b n) 4π a n= = π (a n + b n)(n ) n= Moreover, equalty holds f and only f a n = b n =, for n That s, f and only f p(φ) = a + a cos φ + b sn φ Equvalently, p + p = a Recall that the boundary of a plane compact convex set can be parametrzed by ts support functon p(φ) The angle φ π s the angle between the tangent lne at a gven pont of the boundary and the x axs, and p(φ) s the dstance from ths tangent lne to the orgn Theorem If the boundary = K of a convex set K n the plane s a -curve of length L, then L 4π ρ(s)ds, where ρ(s) s the radus of curvature of, and ds sgnfes arclength measure on Equalty holds f and only f s a crcle Proof Let p(φ) be the support functon of K The followng relatons are well nown (see [4], p 3): L = π p(φ) dφ ρ(φ) = p(φ) + p (φ) The relaton between s and φ s gven by ds = (p(φ) + p (φ))dφ Now the theorem follows drectly by applyng Lemma to the support functon p(φ) n= Note that, by the sopermetrc nequalty, we have proved A ρ(s)ds, where A denotes the area of K Ths s the -dmensonal analogous of Ros nequalty and the dfference ρ(s)ds A was studed n [] We also have an estmaton of the sopermetrc defct: orollary If the boundary = K of a plane convex set K of area A, s a -curve of length L, then L 4π A A e where A e s the (algebrac) area of the doman bounded by the evolute of Equalty holds f and only f s a crcle 3
Proof It s a consequence of Theorem and the fact that ρ(s)ds = (A A e ) (see Theorem, n []) 3 urvature for polygons Gven a plane convex polygon of vertces P, P, P n, we denote by l = P P + the length of ts sdes and by α π the measure of ts external angles Of course we have n = α =, wth < α <, and P P P P + = l l cos(α π) P P "! + " +! P - Defnton Gven a plane convex polygon of vertces P, P, P n, and sdes of lengths l, l,, l n, we defne the radus of curvature at the vertex P by ρ = l + l α π In partcular, the curvature at the vertex P s gven by κ = ρ = l α π + l an expresson that essentally agrees wth the classcal defnton of curvature as the rato of the angle to the length Note also that l = l n Note Another natural defnton of radus of curvature of a polygon (that we wll not use n ths note) s the followng: If P P P + are consecutve vertces of a polygon, the radus of curvature R at P s the radus of the crcumscrbed crcle around the trangle P P P + (see [5]), 4
P - l - P # R "! P + l The relaton between R and ρ s (wth the notaton of the fgure) sn α + sn β ρ = R α + β In partcular, snce Ω = α + β, R tends to ρ when the external angles of the polygon tend to zero Now we shall see (Theorem ) that the radus of curvature, as defned n Defnton, converges to the ordnary radus of curvature of a smooth curve when ths curve s approxmated by polygons We shall consder dyadc approxmatons only for convenence To be precse we gve the followng defnton: Defnton Let γ : [, L] R be the parametrzaton by the arc length of a closed curve of class The n-th dyadc polygon P n assocated to s the polygon gven by the consecutve vertces P (n) = γ(s ), s = L n [, L], =,, 3,, n We shall need the followng lemma: Lemma (auchy s mean value theorem) Let f, g : [a, b] R be contnuous functons, dfferentable on the open nterval (a, b) and let ξ (a, b), such that (f(b) f(a))g (ξ) = (g(b) g(a))f (ξ) (3) If the curve (f(t), g(t)) has curvature >, and we put ξ = a + θh, h = b a, < θ < then lm h θ = 5
Proof Note that f(b) f(a) = f (a)h + f (η ) h, a < η < b, f (ξ) = f (a) + f (η )θh, a < η < ξ, and smlar expressons for g, wth correspondng η 3, η 4 these expressons n auchy s equalty (3) we get Substtutng θ(f (a)g (a) g (a)f (a)) = (f (a)g (η 3 )) f (η )g (a)) + o(), h Tang lmts we obtan the result Theorem Let γ : [, L] R be the parametrzaton by the arc length s of a closed strctly convex curve of class Let ρ (n) be the radus of curvature at the vertex P (n) of the n-th dyadc polygon P n Then, for all ɛ >, there exsts n ɛ N such that for all n > n ɛ, ρ (n) Proof Snce ρ (n) ρ(s (n) ) < ɛ, =,,, n can be approxmated by l(n) + l(n) sn Ω (n) (the sne approxmaton for small angles), t s suffcent to prove that, gven ɛ >, and for n bg enough, we have + l(n) ρ(s (n) sn Ω (n) ) < ɛ, =,,, n (4) where Ω (n) = α (n) π s the exteror angle of the polygon P n at vertex P (n) can be computed usng auchy s mean value theorem The angle Ω (n) In fact, there are ponts η [s (n), s(n) ], η [s (n), s(n) + ], such that y (n) x (n) y (n) x (n) In partcular, we have = y (η ) x (η ), y (n) + y(n) x (n) + x(n) = y (η ) x (η ) γ (η ) γ (η ) = cos Ω Equvalently sn Ω = x (η )y (η ) y (η )x (η ) = x (η ) (y (η ) y (η )) y (η ) (x (η ) x (η )) = (η η ) [x (η )y (τ ) y (η )x (ν )], where τ, ν [η, η ] On the other hand t s clear that the sum + l(n) can be wrtten as ( + l(n) = L (n) x (a ) + y (b ) + ) x (a ) + y (b ) where a, b [s (n), s(n) ], a, b [s (n), s(n) + ] 6
Hence +l(n) x (a ) = +y (b ) + x (a ) +y (b ) L (n) sn Ω (n) (x (η )y (τ ) y (η )x (ν )) η η Let us denote B(, n) = x (a ) + y (b ) + x (a ) + y (b ) (x (η )y (τ ) y (η )x (ν )) Snce s strctly convex, ts curvature s a strctly postve contnuous functon on a compact set (the nterval [, L]) Hence, there s a constant M such that < ρ(s) < M It s clear that B(, n) converges to the radus of curvature In fact we have, for all ɛ >, and for n bg enough, B(, n) ρ(s (n) ) < ɛ/, =,,, n (5) Moreover, snce ρ(s) s bounded, there exsts N > such that B(, n) < N L (n) On the other hand, the fracton converges unformly to η η In fact, by Lemma appled to the functons x(t), y(t), over the ntervals [s, s ] and [s, s + ], and puttng η = s + θ L (n), < θ <, η = s + θ L (n), < θ <, we have L (n) η η = L (n) L (n) +L (n) (θ θ ) <ɛ, =,,, n (6) From nequaltes (5) and (6) we get easly nequalty (4), and theorem s proved 4 A dscrete verson of Theorem In ths secton we shall gve a dscrete verson of Theorem For ths, we shall need the followng result Lemma 3 Let a,, a n R + and let f : (R + ) n R be the functon gven by f(x,, x n ) = a + + a n x x n If x + + x n =, then ( f(x,, x n ) n ) a = 7
Proof Followng the method of Lagrange multplers, we fnd the crtcal ponts of the functon g(x,, x n ) = f(x,, x n ) + λ(x + + x n ) We obtan x = a n = a, =,, n The value of f(x) at ths pont s f(x,, x n )= a x + + a n x n =( a ) a n + +( = = a ) a n = Snce f(x,, x n ) >, and t s not bounded above, the pont a a ( n = a,, n n = a ) s a mnmum Ths completes the proof of the lemma ( n ) a Defnton 3 A convex polygon s called umblcal f the radus of curvature at ts vertces s constant Note that, n ths case, ths constant must be equal to L/π, where L s the length of the polygon Ths fact s easly demonstrated by smply addng the equaltes l + l = α πρ, =,, n, where ρ s the constant radus of curvature Theorem 3 Let L be the length of a convex polygon Then we have L 4π = ρ + ρ + l Equalty holds f and only f the polygon s umblcal Proof By defnton of ρ, the second term of ths nequalty s ρ + ρ + l = ( l + l l + l ) + l + 8π α α + = = 8π = = (l + l + ) α Snce α + + α n =, we can apply Lemma 3 and we obtan Hence 8π = = (l + l + ) α 8π (L) l ρ + ρ + and the nequalty of the theorem s proved 4π L, = 8
By Lemma 3, equalty s attaned when α = (l + l + ) n = (l + l + ) = l + l + L = α πρ L Hence ρ = L π, =,, n and the polygon s umblcal orollary Let A be the area of a convex polygon Then we have A = ρ + ρ + l Proof It s a drect consequence of the sopermetrc nequalty 4πA L Note Note that the term n = l ρ +ρ + can be nterpreted as the area of a rosette composed by sosceles trangles of sdes ρ and angles α π! " l + l + In partcular, equalty n Theorem 3 holds f and only f the rosette s a crcle The defect ρ + ρ + l L 4π = concdes wth the dfference between the area of the rosette and the area of a crcle of radus L/π 9
5 Approxmaton by polygons In ths secton we justfy why Theorem 3 can be vewed as a dscrete verson of Theorem Lemma 4 Let be a closed convex curve of class n the plane Let be the length of the sde P (n) P (n) + of the n-th dyadc polygon P n assocated to Then, for all ɛ >, there exsts n N, such that for all n > n, and for all =,,, n we have n l(n) L < ɛ (7) Proof The proof s standard, usng the mean value theorem Theorem 4 Let γ : [, L] R be the parametrzaton by the arc length s of a closed strctly convex curve of class Let be the length of the sde P (n) P (n) + of the n-th dyadc polygon P n, and let ρ (n) be the radus of curvature at the vertex P (n) Denotng by ρ (n) = ρ(n) + ρ (n) + the arthmetc mean of two consecutve radus of curvature, we have lm n ( n = ρ (n) ) = ρ(s) ds (8) Proof By defnton of Remann ntegral, n order to prove equalty (8) we must only prove Equvalently lm n n = lm n n = Snce ρ (n) ρ(s (n) ) = (ρ (n) ( ρ (n) ρ(s (n) )L(n)) = L (n) ( ) L ρ (n) (n) ρ(s (n) ) = ρ(s(n) ))+(ρ(n) + ρ(s(n) + ))+(ρ(s(n) + ) ρ(s(n) )) and ρ(s) s contnuous, we can assume, by Theorem, that for all ɛ >, there exsts n ɛ N such that for all n > n ɛ, ρ (n) ρ(s (n) ) < ɛ/, =,,, n Note that the rad of curvature ρ (n) are unformly bounded In fact, snce the gven curve s strctly convex, there s a constant M such that < ρ(s) < M So t s clear that, for n bg enough, there exsts N > such that ρ (n) < N
By Lemma 4, gven ɛ >, and for n bg enough, we have From ths t follows easly that and hence n = ɛ N < l(n) L < + ɛ (n) N ɛ < l(n) L ρ (n) (n) ρ(s (n) ) < ɛ, L (n) ( and theorem s proved ) L ρ (n) (n) ρ(s (n) ) < Lɛ, Applyng Theorem 3 to a sequence of dyadc polygons assocated to a closed convex curve, and tang lmts (the length of the polygons converge to the length of the curve) we obtan, by Theorem 4, a dscrete proof of Theorem As ths proof does not use Lemma, and each π-perodc functon p(φ) wth p + p > s the support functon of a convex set, we have gven n fact a geometrcal proof of ths lemma Note that the condton p + p > s not a restrcton snce the addton of a constant to p(φ) leaves nvarant the nequalty of Lemma References [] arlos A Escudero and Agustí Reventós An nterestng property of the evolute Amer Math Monthly, 4(7):63 68, 7 [] Robert Osserman urvature n the eghtes Amer Math Monthly, 97(8):73 756, 99 [3] Antono Ros ompact hypersufaces wth constant scalar curvature and congruence theorem J of Dff Geom, 7:5, 988 [4] Lus A Santaló Integral geometry and geometrc probablty Addson- Wesley Publshng o, Readng, Mass-London-Amsterdam, 976 Wth a foreword by Mar Kac, Encyclopeda of Mathematcs and ts Applcatons, Vol [5] Serge Tabachnov A four vertex theorem for polygons Amer Math Monthly, 7(9):83 833, Departament de Matemàtques Unverstat Autònoma de Barcelona 893 Bellaterra, Barcelona atalunya (Span) jcuf@matuabcat, agust@matuabcat, crodr@matuabcat