A forgotten little chapter on isoperimetric inequalities. On the fraction of a convex and closed plane area lying outside a circle with which it shares a diameter

Rev. Acad. Canar. Cienc., xx(Núms. 1-2), 113-118 (2008) (publicado en septiembre de 2009) AFORGOTTEN LITTLE CHAPTER ON ISOPERIMETRIC INEQUALITIES: ON THE FRACTION OF ACONVEX AND CLOSED PLANE AREA LYING OUTSIDE ACIRCLE WITH WHICH IT SHARES ADIAMETER José M. Pacheco Departamento de Matemáticas. Universidad de Las Palmas de Gran Canaria Abstraet Ofien some inleresting or simply curious points are lefi oul when developing atheory. It seems that one of them is lhe exislence of an upper bound for the fraClion of area of a convex and closed plane area lying outside acircle with which it shares a diameter, a problem stemming from the theory of isoperimelric inequalities. In this paper such a bound is conslructed and shown lO be attained for a particular area. It is also shown lhat convexity is a necessary condilion in order lO avoid lhe whole area Iying outside lhe circle. Key Words: convex plane area, isoperimetric inequality. AMS Classification numbers: 52A 10, 52A38. Introduction Possibly one of lhe oldest extremal problems is lo find aset in Euclidean space with given surface area and enclosing maximum volume. There exist considerable di fferences in lhe lllalhematical trealment of the cases n=2and general n, as shown in the c1assical reference [2], where the general setting directly invites the reader to the reallll of geolllelric measure theory. These problems pervade malhematical activity and lllany mathematicians have dealt with them to different depth degrees. As the mOlivating example for this paper, on reading the book Lifllewood's Miscellany [1] one finds in page 32 lhe following observalion: "An isoperimefrical problem: an area of (greatesl) diameter nol greater than 1is at most -!¡Jr" Littlewood's "greatest diameter" is now better known as fhe diameter d of the plane area, defined as d=sup{diSI(X,nIX, YE boundary}, and without loss of generality, we can suppose in Littlewood's remark the area to be convex and bounded by a conlinuous closed curve. lndeed non-convexity would only amount to reducing the enclosed area, thus enhancing the inequality (see figure \). IPostal address: Campus de Tafira Baja, 35017 LAS PALMAS, Spain. E-mail: [email protected] 113 © Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca universitaria, 2010 Figure 1: Nonconvexily implie8 area 1088. Littlewood's argument is the following (figure in p. 33 of[ 1]), see figure 2: p q o Q '''. Figure 2: A recreation 01 lhe figure in [1J, p. 33. 1Go , area ="2 .1 2(OP- +OQ- )d() Jr """"'."l where OP = p«(), OQ = p«()- -), and OP- +OQ- =PQ- :5 dwm- :5 l. Therefore 2 1" , 1" J[ area =- 12 PQ-d():5 - 12 de =- 2.1 2.1 4 The bound 1 J[ is attained for the unit diameter eirele, and the classieal isoperimetric problem.was to aetually prove that the eirele is the only plane area having this property. Aniee proof based on Fourier expansion teehniques can be found in [4], pp. 181-187, and astandard proof is offered in [2], pp. 104 ff. From an elementary viewpoint there is something eounterintuitive in the geometrical presentation of this inequality beeause, on afirst and erude approximation, the layman eould make avery nai've remark.: Draw an area with unit diameter, and then aeircle sharing adiameter with il. It "seems obvious" that the whole area is eontained in the eirele, so the inequality would be an immediate one (see figure 3, left). 114 © Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca universitaria, 2010 C) Figure 3: The na·,ve idea and lhe counlerexample Of eourse this is erroneolls, as is readily shown by eonsidering an isoseeles triangle KLM whose legs KL and KM are longer than the basis LM. This is aconyex area whose diameter eqllals one of the legs, say KL. Now, let acircle with KL as adiameter be drawn: There is some portion of triangle area in the neighbourhood of comer Mthat lies outside the eircle (see figure 3, right) Anatural question and its answer After the aboye observations, arather natural question immediately arises: /s /here any upper bound lO Ihe frac/ion of area -of a given convex plane arealying ou/side acircle which shares wilh it adiame/er? Figure 4: lIIuslraling lhe procedure. The elassical references [2] and [3] do not mention this topie, and arather thorough Internet search did not proyide direct results, so an attempt to fill this (ittle gap will now be made. In what follows the diameter will be d=1, therefore the area is bounded by711. To start, eonsider aunit diameter eirele (see figure 4to follow the discussion), and let Kand Lbe the endpoints of some diameter thereof. 115 © Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca universitaria, 2010 We shall build aconvex area sharing the diameter KL with the given cirele, and let Xbe any point in the plane. It is evident that both conditions dist(X,K) ~ 1, dist(X,L) ~ J must be satisfied in order that Xbe either an interior or aboundary point of the sought area. Therefore the area is enclosed in the figure defined by two interseeting circle ares centred at Kand Land having unit radius. They define two points Cand Doutside the circle and acurvilinear figure KDLC. Of coursedist(C, D) >1, so KDLC cannot be a eandidate for solving our problem. Therefore we restrict our attention to vertex C (of eourse, its symmetrieal point Dcould be employed as well) and observe that no point in the boundary of the sought area can be more than Iapart from it. Although it seems rather natural to draw the unit radius circle are centred at Cand joining Kand Lto obtain acurvilinear triangle KLC as a more appropriate eandidate, it is elear that the diameter KL does abetter job, and we claim that the mixed triangle KLC mix is the solution to our problem. Ineidentally, the curvilinear triangle KLC is ealled "the Reuleaux triangle" (see Appendix). Ameasure ofhow much area lies outside the circle can now be defined: Simply, it is the ratio between the area outside the circle and the total area just constructed. orea outside the circle l f.l= ~ . total orea It is an easy task to compute the total area ofthe mixed triangleKLC mix : Jr .f3 8Jr - 6.f3 total area(KLC mi ,) = "3 -4 =24 and the fraction outside the circle is: . Jr 5Jr - 6.f3 exterior area(KLC mix )= total area(KLC mix )- - = ---- 824 Therefore, tbe following value is obtained: f.l = 5Jr - 6.f3 == 0.36 8Jr -6.f3 i.e. the maximum fraction of area Iying outside the circle amounts to approximately 36% of the total area. In order to show optimality ofthis result, let us consider adding some area to KLC mix by modifying the boundary curves. This cannot be done by changing the curved side CK (or LC) into another convex curve joining both points for this would imply the existenee of some boundary point at adistance from L(respectively, K) larger than 1, thus contradicting the fact that the sought figure must have unit diameter. Sorne area can be 116 © Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca universitaria, 2010 added below the diameter KL preserving both convexity and unit diameter, but in this case the denominator in the definition of the measure would in crease, thus reducing the value of f.1. The construction also shows that convexity is anecessary condition for the bound to be avalid one. It is enough to observe (see figure 5) that the area outside the circJe is a non-convex figure sharing the unit diameter KL with the circle, but 100% of it lies outside the circle, (and indeed is less than 11r). Figure 5: Convexity is a necessary condition. References [1] Bollobás B(ed) (1997) Littlewood's miscellany, Cambridge University Press, Cambridge UK. [2] Eggleston H(1958) Convexity, Cambridge University Press, Cambridge UK. [3] Mayer A(1935) Der Inhalt der Gleichdicke: Abschatzungen fur ebene Gleichdicke, Mathematische Annalen, 110 (97-127). [4] Nahin P(2006) Dr. Euler 's fabulous formula, Princeton University Press, New York. 117 © Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca universitaria, 2010 Appendix on the Reuleaux triangle The Reuleaux triangle is arather familiar curvilinear triangle (see figure Al) obtained by drawing three cirele ares centred at the vertices of an equilateral triangle with a radius equal to the side, it is indeed convex, and its area is the minimum of all possible figures of constant width having the same diameter d, aresult known as the BlaschkeLebesgue Theorem. These figures share the common length JC xd(a Theorem by Barbier [3]), so the circle and the Reuleaux triangle are extremal curves -in the sense of enclosed areawith this property (see again [3]). Reuleaux triangles have been employed for deeorative purposes (see figure A2) and in technological applieations, such as the Wankel rotary engines. See also Chapter 7 in [2]. Figure A 1: The Reuleaux triangle Figure A2: Reuleaux triangles in Camden Town, London (photograph by lhe aulhor). 118 © Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca universitaria, 2010