Isoperimetric Inequalities in Normed Planes
IIsoperimetric Inequalities in Normed Planes
Rafael S. dos Santos and Marcos Craizer
Abstract.
The classical isoperimetric inequality can be extended toa general normed plane ([2]). In the Euclidean plane, the defect in theisoperimetric inequality can be calculated in terms of the signed areas ofsome singular sets. In this paper we consider normed planes with piece-wise smooth unit balls and the corresponding class of admissible curves.For such an admissible curve, the singular sets are defined as projectionsin the subspaces of symmetric and constant width admissible curves. Inthis context, we obtain some improved isoperimetric inequalities whoseequality hold for symmetric or constant width curves.
Mathematics Subject Classification (2010).
Keywords.
Minkowski geometry; Curves of constant width; Wignercaustic; Isoperimetrix.
1. Introduction
The isoperimetric inequality in the plane is an old problem: It states that L π ≥ A ( γ ) , for any simple convex curve γ , with equality holding only for circles. Theisoperimetric inequality has been extended to an arbitrary normed planeby Busemann. In this case, the curve of a fixed area that minimizes lengthis not the unit circle, but the dual unit circle, also called isoperimetrix ([2],[9],[10],[13],[14]). By duality, for a fixed area, the unit circle is the curvethat minimizes the dual length. The authors wants to thank CNPq and CAPES (financial code 001) for financial supportduring the preparation of this manuscript. This manuscript arises from the Ph.D. disser-tation of the first author under the supervision of the second author ([12]).E-mail of the corresponding author: [email protected]. a r X i v : . [ m a t h . M G ] O c t R.Segadas and M.CraizerNormed planes with smooth strictly convex unit circles were considered in[6] and normed planes with polygonal unit circles were considered in [7]. Inboth cases, the notion of dual length of a convex curve can be extended toa (signed) dual length in a class C of curves, whose elements are called ad-missible curves. In the strictly convex case, the class C of admissible curvesconsists of the smooth Lagrangian curves of degree 1, while for the polygonalcase, the admissible curves are polygons with sides parallel to the unit circle.Consider the subspaces C cw ⊂ C of symmetric curves and C sym ⊂ C of con-stant width curves, both with zero dual length. Then the mixed area definesan inner product in C such that C cw and C sym become orthogonal. In thispaper we generalize these constructions to normed planes whose unit ball ispiecewise smooth.For a normed plane with piecewise unit circle, we define a class C of ad-missible curves and denote by C cw ⊂ C the class of constant width curveswith zero dual length. Then the orthogonal projection π cw ( γ ) of γ ∈ C on C cw with respect to the signed inner product given by the mixed area is awell-known object, the Wigner caustic , also called area evolute or mid-pointparallel tangent line of γ ([4],[5],[8]).The orthogonal complement of C cw in C can be orthogonally decomposedin C sym , the space of symmetric curves with zero dual length and the one-dimensional subspace U consisting of multiples of the unit circle. In the eu-clidean case, the orthogonal projection π sym ( γ ) of γ ∈ C on C sym was calledthe constant with measure set of γ in [16].Based on the above decompositions, we prove a formula for the isoperimetricdefect related to the signed areas of π cw ( γ ) and π sym ( γ ). Our approach,besides providing another proof of the isoperimetric inequality of [2], alsogives rise to some improved isometric inequalities whose equalities hold onlyfor symmetric or constant width curves. These results generalize the resultsof [15] and [16] in the euclidean plane. We also give a proof of Lhuilier’sinequality, taking advantage of the fact that our proof holds for smooth unitcircles as well as for polygonal unit circles.The paper is organized as follows: In section 2 we define the admissible curvesassociated with the unitary ball of a normed plane. In section 3 we discussmixed areas. In section 4 we decompose the space of admissible curves withzero dual length in two orthogonal subspaces and discuss the properties of theprojections of a curve in these subspaces. In section 5 we prove the improvedisoperimetric inequalities, which are the main results of the paper.soperimetric Inequalities in Normed Planes 3
2. Unit Ball and Admissible Curves
Consider a normed plane with unit ball U . We shall assume that the boundaryof U , u = ∂U , is piecewise smooth. More precisely, we shall assume that u = (cid:83) ≤ i ≤ n u i , where u i are smooth arcs with u i + n = − u i . We shall alsoassume that each u i is either a smooth strictly convex arc or a straightsegment.The dual ball U ∗ is defined as U ∗ = { f ∈ ( R ) ∗ | || f || ≤ } , where || f || = sup {| f ( x ) | | x ∈ U } . Each functional f ∈ ( R ) ∗ can be repre-sented by a vector v ∈ R by the relation f ( x ) = [ x, v ] , x ∈ R , (2.1)where [ a, b ] denotes the determinant of the 2 × a, b ∈ R . We shall represent U ∗ by V under this identification.Given v ∈ R , let L v denote the support line of U parallel to v , and assume L v touches U in u ∈ ∂U . Then v ∈ V if and only if [ u, v ] = 1. Thus if u belongs to a smooth arc, the corresponding v also belongs to a smooth arc.If u belongs to a straight segment, then v is constant. Finally if u is a vertex,then v describes a segment. An example of a piecewise smooth unit ball U and its dual V can be seen in Figure 1. Figure 1.
An example of a piecewise smooth unit ball andits dual.
Consider a parameterization u ( t ) of u such that u (cid:48) ( t ) (cid:54) = 0, for any t ∈ [ t i , t i +1 ],where u ( t i ) are the vertices of u . There are 2 types of intervals [ t i , t i +1 ]: Thesmooth intervals, where u and v are strictly convex, and the linear intervals,where u is a straight segment and v is constant. At smooth intervals we shall R.Segadas and M.Craizerassume also that [ u (cid:48) ( t ) , u (cid:48)(cid:48) ( t )] (cid:54) = 0. For any interval t i ≤ t ≤ t i +1 , we canwrite v ( t ) = u (cid:48) ( t )[ u ( t ) , u (cid:48) ( t )] . (2.2)In fact, the functional associated with v ( t ) under the identification (2.1) iszero in the direction u (cid:48) ( t ) and is one at u ( t ). Thus it belongs to the dual unitcircle at direction u (cid:48) ( t ). Since the straight lines are not included in Equation(2.2), this formula parameterizes only a part of the dual circle, but in factthis part is all that we need in the paper. Example . Consider the unit ball shown in Figure 1. Then a parameteriza-tion for the unit circle is given by u ( t ) = (1 − t, t ) , ≤ t ≤ , (cos( π t ) , sin( π t )) , ≤ t ≤ , ( t − , − t ) , ≤ t ≤ , (cos( π t ) , sin( π t )) , ≤ t ≤ , The corresponding points in the dual circle are given by v ( t ) = ( − , , ≤ t ≤ , ( − sin( π t ) , cos( π t )) , ≤ t ≤ , (1 , − , ≤ t ≤ , ( − sin( π t ) , cos( π t )) , ≤ t ≤ . Note that this parameterization does not include the straight lines of the dualball.
Denote by C = C ( U ) the class of piecewise smooth closed curves γ parame-terized by 0 ≤ t ≤ T , γ (0) = γ (2 T ) such that, for t i ≤ t ≤ t i +1 , γ (cid:48) ( t ) = r ( t ) u (cid:48) ( t ) , (2.3)for some scalar function r ( t ). We remark that when u is smooth, the class C includes all the convex smooth curves ([6]), and when u is polygonal, theclass C consists of all polygons with sides parallel to those of u ([7]).The scalar r ( t ) = r γ ( t ) is called the curvature radius of γ at γ ( t ) ([1],[11]). Upto sign, the curvature radius is independent of the choice of the parametriza-tions of γ and u . At an interval where u is a straight segment, Equation (2.3)implies that γ is also a straight segment and r ( t ) is the rate between thelengths of the γ and u segments. At a smooth interval, we have that[ γ (cid:48) ( t ) , γ (cid:48)(cid:48) ( t )] = r ( t ) [ u (cid:48) , u (cid:48)(cid:48) ]( t ) , which implies that γ has no inflection points. It is clear from Equation (2.3)that, up to a translation, we can recover γ from r . It is easy to see from thecurvature radius whether or not the curve is convex, as next lemma shows: Lemma 2.1.
The curve γ ∈ C is convex if and only if r ( γ ) does not changesign. soperimetric Inequalities in Normed Planes 5 Proof.
Denote γ (cid:48) ( t + ) = lim s → t + γ (cid:48) ( t ) and γ (cid:48) ( t − ) = lim s → t − γ (cid:48) ( s ). If r changessign at a point t of a smooth interval, then γ (cid:48) ( t +0 ) and γ (cid:48) ( t − ) are pointingin opposite directions and so the curve is not convex at this point. Similarly,if r changes sign at a vertex t , then γ (cid:48) ( t +0 ) and γ (cid:48) ( t − ) make an angle biggerthan π , and again the curve is not convex at this point.Conversely, if r does not change sign, the tangents at vertices are makingangles strictly smaller than π and so the curve is locally convex at the vertices.Since the curve has no inflection points, it is also locally convex at the smootharcs. Finally, since the index of γ is ±
1, the curve γ is necessarily convex. (cid:3) Corollary 2.2.
Given γ ∈ C , there exists a constant K > such that γ + Ku is convex.Proof. The curvature radius of γ + Ku is r ( t ) + K . If we choose K ≥− min { r ( t ) } , we obtain a convex curve. (cid:3) Given γ ∈ C , from Equations (2.2) and (2.3), we can write γ (cid:48) ( t ) = r ( t )[ u, u (cid:48) ]( t ) v ( t ) , for any t in a smooth or linear interval. The (signed) dual length of γ ∈ C isdefined by L ∗ ( γ ) = (cid:90) T r ( t )[ u, u (cid:48) ]( t ) dt. (2.4)When γ is convex, L ∗ ( γ ) coincides with the dual length of γ . Example . Let γ ( t ) = (2 − t, t ) , ≤ t ≤ , √
15 cos ( π t )+1 (16 cos( π t ) , sin( π t )) + (1 , , ≤ t ≤ , ( −
11 + 4 t, − t ) , ≤ t ≤ , √
15 sin ( π t )+1 (cos( π t ) ,
16 sin( π t )) + (1 , , ≤ t ≤ , (see Figure 2). Figure 2.
A curve in C (Example 2). R.Segadas and M.CraizerThe curvature radius is given by r ( t ) = , ≤ t ≤ , √ (15 cos ( π t )+1) , ≤ t ≤ , , ≤ t ≤ , √ (15 sin ( π t )+1) , ≤ t ≤ , We can calculate the dual length of γ by equations (2.4) to obtain L ∗ ( γ ) ≈ . .
3. Mixed areas
Consider two curves γ , γ ∈ C . Then the mixed area is defined by A ( γ , γ ) = 12 (cid:90) T [ γ , γ (cid:48) ]( t ) dt. The signed area of γ ∈ C is defined as A ( γ ) = A ( γ, γ ). If γ is convex, A ( γ ) isthe area of the region bounded by γ . Lemma 3.1. A ( γ , γ ) is a symmetric bilinear map in C . Moreover, L ∗ ( γ ) = 2 A ( u, γ ) . (3.1) Proof.
Since [ γ , γ ] (cid:48) = [ γ (cid:48) , γ ] + [ γ , γ (cid:48) ] we obtain (cid:90) T [ γ , γ (cid:48) ]( t ) dt = (cid:90) T [ γ , γ (cid:48) ]( t ) dt. For the second assertion observe that L ∗ ( γ ) = (cid:90) T r ( t )[ u, u (cid:48) ]( t ) dt = (cid:90) T [ u, γ (cid:48) ]( t ) dt = 2 A ( u, γ ) , thus proving the lemma. (cid:3) The Minkowski inequality states that for convex curves γ ∈ C , A ( γ , γ ) ≥ A ( γ ) A ( γ ) , with equality if and only if γ is a multiple of γ . Lemma 3.2.
For any γ ∈ C , L ∗ ( γ ) ≥ A ( γ ) A ( u ) , with equality if and only if γ is a multiple u . soperimetric Inequalities in Normed Planes 7 Proof.
Since γ ∈ C , by Corollary 2.2 there exists a constant K > γ + Ku are convex. Then, by Minkowski inequality, A ( γ + Ku, u ) ≥ A ( γ + Ku ) A ( u ) . Since A ( γ + Ku ) = A ( γ )+ K A ( u )+2 KA ( γ, u ) and A ( γ + Ku, u ) = A ( γ, u )+ KA ( u ) , we conclude that A ( γ, u ) ≥ A ( γ ) A ( u ) , thus proving the lemma. (cid:3) We say that γ ∈ C is of constant width if[ γ, v ]( t + T ) + [ γ, v ]( t ) = c, for some constant c . Lemma 3.3.
For constant width curves, we have that L ∗ ( γ ) = 2 cA ( U ) and so necessarily c = w γ .Proof. Since2 A ( γ, u ) = (cid:90) T [ γ, u (cid:48) ] dt = (cid:90) T [ u, u (cid:48) ] ([ γ, v ]( t ) + [ γ ( t + T ) , v ( t + T )]) dt we conclude that L ∗ ( γ ) = 2 cA ( U ). (cid:3) Corollary 3.4.
A curve γ ∈ C with zero dual length is of constant width ifand only if [ γ, v ]( t + T ) + [ γ, v ]( t ) = 0 .
4. The subspaces of constant width and symmetriccurves
In this section we consider the mixed area as a signed inner product in C . C Denote by U the 1-dimensional subspace of C consisting of the constant mul-tiples of the unit ball, by C sym ⊂ C the subspace consisting of symmetriccurves with respect to the origin with zero dual length and by C cw ⊂ C thesubspace consisting of constant width curves, also with zero dual length. ThenEquation (3.1) implies that both C sym and C cw are orthogonal to U . Lemma 4.1.
The subspaces C sym and C cw are orthogonal. R.Segadas and M.Craizer
Proof. If γ ∈ C cw and γ ∈ C sym , we have that[ γ ( t + T ) , v ( t + t )] + [ γ ( t ) , v ( t )] = 0 , γ ( t + T ) = − γ ( t ) . Thus A ( γ , γ ) = (cid:90) T [ γ , γ (cid:48) ]( t ) dt + (cid:90) T [ γ , γ (cid:48) ]( t + T ) dt, = (cid:90) T r ( t )[ u, u (cid:48) ]( t ) ([ γ ( t ) , v ( t )] − [ γ ( t + T ) , v ( t )]) dt, = (cid:90) T r ( t )[ u, u (cid:48) ]( t ) ([ γ ( t ) , v ( t )] + [ γ ( t + T ) , v ( t + T )]) dt, which proves that A ( γ , γ ) = 0. (cid:3) For γ ∈ C , let γ ( t ) = 12 ( γ ( t ) + γ ( t + T ))be the Wigner caustic of γ . Observe that[ γ ( t ) , v ( t )] + [ γ ( t + T ) , v ( t + T )] = [ γ ( t ) , v ( t )] − [ γ ( t ) , v ( t )] = 0 , which implies that γ has constant width 0, i.e., γ ∈ C cw . Moreover γ ( t ) − γ ( t ) = 12 ( γ ( t ) − γ ( t + T ))is symmetric, which means ( γ − γ ) ∈ C sym ⊕ U . We conclude that π cw ( γ ) = 12 ( γ ( t ) + γ ( t + T )) . In other words, the Wigner caustic of γ coincides with the orthogonal pro-jection of γ on C cw .Now let γ ( t ) = 12 ( γ ( t ) − γ ( t + T ) − w γ u ( t )) , where w γ = L ∗ ( γ ) A U is the mean width of γ . In the euclidean case, γ was called the constant widthmeasure set of γ in [16]. Is is clear that γ is symmetric and L ∗ ( γ ) = L ∗ ( γ ) − (cid:90) T w γ [ u, u (cid:48) ]( t ) dt = 0 , thus proving that γ ∈ C sym . Moreover γ ( t ) − γ ( t ) = γ ( t ) + 12 w γ u ( t )has constant width, which means ( γ − γ ) ∈ C cw ⊕ U . We conclude that π sym ( γ ) = 12 ( γ ( t ) − γ ( t + T ) − w γ u ( t )) . soperimetric Inequalities in Normed Planes 9For future reference, we remark that γ = π cw ( γ ) + π sym ( γ ) + w γ u. (4.1) Proposition 4.2.
We have that A ( π cw ( γ )) ≤ with equality if and only if γ is symmetric and A ( π sym ( γ )) ≤ with equality if and only if γ is of constantwidth.Proof. The Proposition follows immediately from the fact that L ∗ ( π cw ( γ )) = L ∗ ( π sym ( γ )) = 0and Lemma 3.2. (cid:3) Example . The π cw ( γ ) and π sym ( γ ) of the curve γ of Example 2 can be seenin Figure 3. Figure 3.
The projections π cw ( γ ) and π sym ( γ ) of the curve γ of Example 2. Their areas are approximately − .
33 and − .
48, respectively.
5. Improved Isoperimetric Inequalities
Let γ ∈ C be a convex curve. Then L ∗ ( γ )4 A ( U ) = A γ − A ( π cw ( γ )) − A ( π sym ( γ )) . Proof.
Consider the orthogonal decomposition of γ given by Equation (4.1).Taking into account that W C is T -periodic, we obtain A ( γ ) = 2 A ( π cw ( γ )) + A ( π sym ( γ )) + w γ A ( U ) . We conclude that A ( γ ) − A ( π cw ( γ )) − A ( π sym ( γ )) = L ∗ ( γ )4 A ( U ) , thus proving the proposition. (cid:3) The next corollary gives us two new improved isoperimetric inequalities:
Corollary 5.2.
Let γ ∈ C be a convex curve. The following improved isoperimetric inequality holds: L ∗ ( γ )4 A ( U ) ≥ A ( γ ) − A ( π sym ( γ )) , with equality if and only if γ is symmetric. The following improved isoperimetric inequality holds: L ∗ ( γ )4 A ( U ) ≥ A ( γ ) − A ( π cw ( γ )) , with equality if and only if γ is of constant width. Next corollary recovers the isoperimetric inequality of Busemann in a morerestricted case. In fact, Busemann’s inequality holds for any convex curve γ and any normed plane (see [2]). Corollary 5.3.
Let γ ∈ C be a convex curve. Then L ∗ ( γ )4 A ( U ) ≥ A ( γ ) , (5.1) and equality holds if and only if γ is a multiple of the unit ball. In this section we prove Lhuilier’s inequality, taking advantage of the factthat our result holds not only for smooth unit circles but also for polygonalunit circles.Consider a convex poligon K and let K be the polygon which is circum-scribed about the unit circle and whose sides are respectively parallel to thesides of K . Let L ∗ ( K ) denotes the dual length of K , A ( K ) the area enclosedby K and A ( K ) the area enclosed by K . Then Lhuilier’s theorem ([3]) statesthat L ∗ ( K )4 A ( K ) ≥ A ( K ) , with equality if and only if K is a constant multiple of K .We shall now proof a weaker version of Lhuilier’s inequality, namely L ∗ ( K )4 A ( K ) ≥ A ( K ) , (5.2)where K = K ∩ ( − K ) is the symmetrization of K , with equality if and onlyif K is a constant multiple of K . This inequality coincides with Lhuilier’s in-equality if K has parallel opposite sides. To prove the general case of Lhuilier’ssoperimetric Inequalities in Normed Planes 11inequality, we need to develop the results of this paper in the more generalcontext of a non-symmetric unit ball.To prove Inequality (5.2), consider the normed plane with unit ball K .Observe that the dual length L ∗ ( K ) with respect to original normed plane isthe same as the dual length of K with respect to the normed plane with unitball K . Then by Inequality (5.1) applied to K ∈ C ( K ), i.e., in the normedplane whose unit ball is K , we obtain Inequality (5.2). References [1] V.Balestro, H.Martini and E.Shonoda:
Concepts of curvature in normed planes ,Exp.Math., 37(4), 347-381, 2019.[2] H.Busemann:
The Isoperimetric Problem in the Minkowski Plane , Amer. J.Math., 69(4), 863-871, 1947.[3] G.D.Chakerian:
The Isoperimetric Problem in the Minkowski Plane , Amer.Math. Monthly, 67(10), 1002-1004, 1960.[4] M.Craizer:
Iterates of involutes of constant width curves in the Minkowski plane ,Beitr¨age zur Algebra und Geometrie, 55(2), 479-496, 2014.[5] M.Craizer and H.Martini:
Involutes of constant width polygons in the Minkowskiplane , Ars Math.Comtemp., 11(1), 107-125, 2016.[6] M.Craizer, R.C.Teixeira, and V.Balestro:
Closed cycloids in a normed plane ,Monatshefte f¨ur Math., 185(1), 43-60, 2018.[7] M.Craizer, R.C.Teixeira, and V.Balestro:
Discrete cycloids from convex sym-metric polygons , Disc.Comp.Geom., 60(4), 859-884, 2018.[8] Giblin,P.J.:
Affinely invariant symmetry sets . Geometry and Topology of Caus-tics (Caustics 06), Banach Center Publications , 71-84 (2008).[9] H. Guggenheimer: On plane Minkowski geometry , Geom.Dedicata, 12(4), 371-381, 1982.[10] H.Martini, Z. Mustafaev:
On isoperimetric inequalities in Minkowski Spaces ,J.Inequal.Appl. 2010, 697954, 2010.[11] C.M.Petty:
On the geometry of the Minkowski plane , Riv. Mat. Univ. Parma,6, 269-292, 1955.[12] R.S.dos Santos:
C´austicas de Wigner e conjuntos de medida de largura con-stante em planos normados com bolas unit´arias suaves ou poligonais , Ph.D.dissertation, in portuguese, 2019.[13] G.Strang:
Maximum area with minkowski measure of perimeter , Proc. RoyalSoc. Edinburgh 138A, 189-199, 2008.[14] A.C.Thompson:
Minkowski Geometry . Encyclopedia of Mathematics and itsApplications, 63, Cambridge University Press, 1996.[15] Zwierzynski, M.
The improved isoperimetric inequality and the Wigner causticof planar ovals , J.Math.Anal.App., 442(2), 726-739, 2016.[16] Zwierzynski, M.
The constant width measure set, the spherical measure set andisoperimetric equalities for planar ovals , arXiv:1605.02930, 2016.
Rafael S. dos SantosDepartamento de Matem´atica- PUC-RioRio de Janeiro, RJ, Brasile-mail: [email protected]