A SHARP LOWER BOUND ON THE POLYGONAL ISOPERIMETRIC DEFICIT

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1 A SHARP LOWER BOUND ON THE POLYGONAL ISOPERIMETRIC DEFICIT EMANUEL INDREI Abstract. It is shown that the isoperimetric deficit of a convex polygon P admits a lower bound in terms of the variance of the radii of P, the area of P, and the variance of the barycentric angles of P. The proof involves circulant matrix theory and a Taylor expansion of the deficit on a compact manifold. 1. Introduction The polygonal isoperimetric inequality states that if n 3 and P is an n-gon with area P and perimeter LP, then the deficit is nonnegative, δp := L P 4n tan π P 0, n and uniquely minimized when P is convex and regular. A sharp stability result for this classical inequality has recently been obtained in [IN15] via a novel approach involving a functional minimization problem on a compact manifold and the spectral theory for circulant matrices. The heart of the matter is a quantitative polygonal isoperimetric inequality for convex polygons which states that 1.1 σ sp + σ rp δp, where σ sp is the variance of the side lengths of P and σ rp is the variance of its radii i.e. the distances between the vertices and their barycenter. The starting point of the proof is the following inequality [FRS85, pg. 35] which holds for any n-gon: 1. 8n sin π n σ rp nsp 4n tan π P, n where SP is the sum of the squares of the side lengths of P. Since n σsp = nsp L P, it follows that 1. is equivalent to 1.3 8n sin π n σ rp δp + n σ sp. In order to establish 1.1, it is shown in [IN15] that 1.4 σ sp δp PIRE Postdoctoral Fellow. 1

2 E. INDREI whenever P is a convex n-gon; thereafter, a more general stability result e.g. valid for simple n-gons is deduced via a version of the Erdős-Nagy theorem which states that a polygon may be convexified in a finite number of flips while keeping the perimeter invariant. The method of proof of 1. given in [FRS85] is based on a polygonal Fourier decomposition, whereas the technique in [IN15] is based on a Taylor expansion of the deficit in a suitable sense. It is natural to wonder whether one can directly deduce 1.1 via the method in [IN15] without relying on [FRS85]. A positive answer is given in this paper. In fact, a new inequality is established which combined with 1.4 improves 1.1. Let σap denote the variance of the barycentric angles of P i.e. the angles generated by the vertices and barycenter of the set of vertices of P, see. Then the following is true. Theorem 1.1. Let n 3 and P be a convex n-gon. There exists c n > 0 such that and the exponent on the deficit is sharp. c n δp σ rp + P σ ap, This result directly combines with 1.4 and yields: Corollary 1.. Let n 3 and P be a convex n-gon. There exists c n > 0 such that c n δp σ sp + σ rp + P σ ap. Remark 1.3. The theorem holds for a more general class of polygons. The only requirement in the proof is that the barycentric angles of P sum to π. Remark 1.4. An inequality of the form σ ap c n δp cannot hold in general. One can see this by a simple scaling consideration: let P be a convex polygon and P α be the convex polygon obtained by dilating the radii of P by α > 0. Then δp α = α δp, but σ ap α = σ ap. Quantitative polygonal isoperimetric inequalities turn out to be useful tools in geometric problems. For instance 1.1 was recently utilized in [CM14] to improve a result of Hales which showed up in his proof of the honeycomb conjecture [Hal01]. This was achieved by showing that the notion of asymmetry in 1.1 directly controls the Hausdorff distance between P and a specific regular polygon. Moreover, [IN15] has also been employed in [CN15] to prove a quantitative version of a Faber-Krahn inequality for the Cheeger constant of n-gons obtained in [BF15]. Related stability results for the isotropic, anisotropic, and relative isoperimetric inequalities have been obtained in [FMP08, FMP10, FI13], respectively.

3 Acknowledgements The author is pleased to acknowledge support from NSF Grants OISE PIRE, DMS , and DMS administered by the Center for Nonlinear Analysis at Carnegie Mellon University. Moreover, the excellent research environment provided by the Hausdorff Research Institute for Mathematics and the Rheinische Friedrich- Wilhelms-Universität Bonn is kindly acknowledged. Lastly, the author wishes to thank an anonymous referee for providing useful feedback on a preliminary version of this paper.. Preliminaries Let n 3 and P R be an n-gon generated by the set of vertices A 1, A,..., A n } R whose center of mass O is taken to be the origin. For i 1,,..., n}, the i-th side length of P, denoted by l i := A i A i+1, is the length of the vector A i A i+1 which connects A i to A i+1, where A i = A j if and only if i = j mod n; with this notation in mind, r i := OA i } n is the set of radii. Furthermore, x i is the angle between the vectors OA i and OA i+1 and the set x i } n comprises the barycentric angles of P. The circulant matrix method introduced in [IN15] is based on the idea that a large class of polygons can be viewed as points in R n satisfying some constraints. More precisely, consider } M := x; r R n : x i, r i 0,.1,.,.3 hold, 3 where.1 x i = π,..3 r i = n. i 1 r i cos r i sin i 1 x k = 0, x k = 0. Note that M is a compact n 4 dimensional manifold and each point x; r M represents a polygon centered at the origin with barycentric angles x and radii r; therefore, it is appropriate to name such objects polygonal manifolds. Indeed, a point O is the barycenter of the set of vertices of P if and only if OA i = 0,

4 4 E. INDREI which is equivalent to saying that the projections of OA i onto OA 1 and OA 1 vanish; in other words, x; r satisfies.3. Furthermore,.1 is satisfied by all convex polygons also many nonconvex ones and. is a convenient technical assumption which derives from scaling considerations. Note that the convex regular n-gon corresponds to the point x ; r = π,..., π; 1,..., 1. With this in mind, the variance of n n the interior angles and radii of P are represented, respectively, by the quantities σ ap = σ ax; r := 1 n x i 1 x n i, σrp = σrx; r := 1 ri 1 r n n i. Moreover, in x; r coordinates, the deficit is given by the formula δp = δx; r := r i+1 + ri 1/ r i+1 r i cos x i n tan π n r i r i+1 sin x i. 3. Proof of Theorem 1.1 By a simple reduction argument, it suffices to prove the inequality on M: let P be a convex n-gon and note that it is represented by x; r R n, where x R n denotes its interior angles and r R n its radii. Convexity implies.1, and.3 follows from the definition of barycenter. If r i = s n, consider by a slight abuse of notation the polygon P s = x; n r obtained by scaling s the radii of P. Evidently σap s = σap, P s = n/s P, σrp s = n/s σrp, δp s = n/s δp. Hence if the inequality stated in the theorem holds for P s M, then it also holds for P. Now let φx; r : = n P σa + σr = 1 r i r i+1 sin x i n and note that it suffices to show x i n x i + n 3.1 φx; r c δx; r ri n, r i for all x; r M. The polygonal isoperimetric inequality implies δx; r 0 for every x; r M with δx; r = 0 if and only if x; r = z := x ; r. Since M is compact

5 and δ is continuous it follows that for µ > 0, inf δ > 0, M\B µz and so 3.1 follows easily on M \ B µ z. Thus it suffices to prove 3.1 for some neighborhood B µ of the point z. Direct calculations imply recall that the notation is periodic mod n 3. Dφz := D x φz, D r φz = 0, D xk x l φz = D rk r l φz = nn 1 sin π, n k = l, n sin π, n k l, n 1, k = l,, k l, and D rk x l φz = 0. Thus by letting Φ := D φz it follows that n sin π Φ = C 0 n n n, 0 n n C where 0 n n is the n n zero matrix and n n n C = n n 1 n n. 5 Moreover, Dδz is given by Dxk δz = n tan π n, D rk δz = 0; hence,.1 implies Dδz, x x ; r r = D x δz, x x + D r δz, r r = n tan π 3.3 x i x i = 0. n Since φz = δz = 0, by utilizing 3., 3.3, and performing a Taylor expansion, it follows that for z close enough to z,

6 6 E. INDREI φz = 1 D φz z z, z z and i,j, D ijk φ1 θ z z + θ z zz z i z z j z z k, δz = 1 D δz z z, z z i,j, D ijk δ1 τ z z + τ z zz z i z z j z z k, for some θ z, τ z 0, 1. Furthermore, since in a neighborhood of z, φ and δ are C 3 and M is compact, there exists C > 0 such that 1 D ijk φ1 θ z z + θ z zz z i z z j z z k C z z 3, i,j, i,j, D ijk δ1 θ z z + θ z zz z i z z j z z k C z z 3, for z M sufficiently close to z. Thus there exists C > 0 for which φz 1 D φz z z, z z C z z 3, δz 1 D δz z z, z z C z z 3, in a neighborhood of z. In particular, there exists η = ηn > 0 such that 3.6 φz 1 Φ z z + C z z 3 for all z B η z. By the results of [IN15, 3.6 ], it follows that inf w S H D δz w, w =: σ > 0, 1 1 In fact, something stronger is proved: namely that inf w SH D fz w, w =: σ > 0 where f is an explicit function for which D f D δ. This is achieved via the spectral theory for circulant matrices and an analysis involving the tangent space of M at z and the identification of a suitable coordinate system in which calculations can be performed efficiently. The barycentric condition.3 built into the definition of M comes up in this analysis.

7 where H is the tangent space of M at z and S H is the unit sphere in H with center z. Moreover by continuity there exists a neighborhood U R n of S H such that D δz w, w σ, for all w U. Note that z z z z U for z M sufficiently close to z. Hence there exists µ = µη, σ 0, η] such that D δz z z, z z σ z z for z B µ z. In particular, for µ := minµ, σ 8C } and z B µz, δz 1 4 D δz z z, z z ; thus, recalling 3.6, 1 φz σ Φ + C σ z z D δz z z, z z c n δz, where c n := 4 Φ σ + 8C µ. To achieve the second part of the theorem, it suffices to σ prove the existence of c > 0 such that 3.7 Φ x; r, x; r c x; r, for x; r Z := x; r : x i = 0, } r i = 0. Indeed, if 3.7 holds, let ω : [0, ] [0, ] be any modulus of continuity i.e. ω0+ = 0 such that φz c n ωδz. Then for z M close to z, 3.5 implies δz c 0 z z, for some c 0 > 0. Moreover, z z Z since z M, and by combining 3.4 with 3.7 it follows that 3.8 δz c 0 z z c 1 Φz z, z z c φz cωδz, for some c > 0 provided z is close to z ; however, since δz 0 as z z and δz > 0 for z z, 3.8 leads to a contradiction if lim inf t 0 + ωt t Thus the lim inf is strictly greater than zero and this implies ω is at most linear at zero. To verify 3.7, note first that C is a real, symmetric, circulant matrix generated = 0. 7

8 8 E. INDREI by the vector n 1, 1,..., 1. A calculation shows that the eigenvalues of C, say λ k, are given by 3.9 λ 0 = 0 and λ k = n for k = 1,..., n 1. Moreover, let v 0 := 1,..., 1, and for l 1,..., n } define v l 1 := 1, cos πl 4πl, cos n n n v l := 0, sin πl n πln 1,..., cos,. 4πl πln 1, sin,..., sin n n One can readily check that v k is an eigenvector of C corresponding to the eigenvalue λ k, and that the set v 0, v 1,..., v n 1 } forms a real orthogonal basis of R n see e.g. Proposition.1 in [IN15]. For k = 1,,..., n, define b k := v k 1 ; 0,..., 0 R n and b k := 0,..., 0; v k n 1 R n for k = n + 1,..., n. Since the set b k } n forms a real orthogonal basis of R n, given x; r R n there exist unique coefficients α k R such that Thus, by utilizing 3.9 it follows that Φx; r, x; r = Furthermore, if x; r Z, k,k =1 = n sin π n = n sin π n x; r = α k α k Φb k, b k α k b k. α kλ k 1 b k + αk b k + n k= α 1 = x; r, b 1 b 1 = k=n+1 k=n+ x i = 0, α n+1 = x; r, b n+1 b 1 = r i = 0; α k b k. α kλ k n 1 b k

9 9 hence, Φ x; r, x; r = n sin π n and this concludes the proof. n αk b k + n α k b k, References k=n+1 α k b k [BF15] M. Bucur and I. Fragala. A Faber-Krahn inequality for the Cheeger constant of N-gons. J. Geom. Anal., DOI /s [CM14] M. Caroccia and F. Maggi. A sharp quantitative version of Hales isoperimetric honeycomb theorem. arxiv: [CN15] M. Caroccia and R. Neumayer. A note on the stability of the Cheeger constant of N-gons. J. Convex Anal., to appear 015. [FI13] A. Figalli and E. Indrei. A sharp stability result for the relative isoperimetric inequality inside convex cones. J. Geom. Anal., 3: , 013. [FMP08] N. Fusco, F. Maggi, and A. Pratelli. The sharp quantitative isoperimetric inequality. Ann. of Math., 1683: , 008. [FMP10] A. Figalli, F. Maggi, and A. Pratelli. A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math., 181:167 11, 010. [FRS85] J. Chris Fisher, D. Ruoff, and J. Shilleto. Perpendicular polygons. Amer. Math. Monthly, 91:3 37, [Hal01] T. C. Hales. The honeycomb conjecture. Discrete Comput. Geom., 51:1, 001. [IN15] E. Indrei and L. Nurbekyan. On the stability of the polygonal isoperimetric inequality. Advances in Mathematics, 76:6 86, 015. Emanuel Indrei Center for Nonlinear Analysis Carnegie Mellon University Pittsburgh, PA 1513, USA egi@cmu.edu