Isoperimetric stability of boundary barycenters in the plane

Consider an open domain D on the plane, whose isoperimetric deﬁcit is smaller than 1. This note shows that the diﬀerence between the barycenter of D and the barycenter of its boundary is bounded above by a constant times the isoperimetric deﬁcit to the power 1/4. This power can be improved to 1/2, when D is furthermore assumed to be a convex domain, in any Euclidean space of dimension larger than 2.


Introduction
Consider a plane simple closed (or Jordan) curve C of length L ă `8, bounding an open domain D of area A. The usual isoperimetric inequality asserts that L 2 ě 4πA (1) and that the equality is attained if and only if D is a disk.The field of isoperimetric stability investigates what can be said about D when (1) is close to an equality, under an appropriate renormalisation.Recently there has been a lot of progress in this direction, see for instance the lecture notes of Fusco [5] and the references therein.Define ρ a A{π and the barycenter bpDq of D by There are several ways to measure how far D is from BpbpDq, ρq, the disk centered at bpDq of radius ρ, when the isoperimetric deficit is small.Here, we are interested in the difference between bpDq and the barycenter bpCq of the boundary C, defined by where σ is the one-dimensional Hausdorff measure (so that in particular σpCq " L).
Of course when dpDq " 0, we have bpCq " bpDq " bpBpbpDq, ρqq.It seems that the isoperimetric stability of the boundary barycenter has not been studied before.Our primary motivation comes from an illustrative example on the plane in [3], which investigates certain domain-valued stochastic evolutions associated by duality with elliptic diffusions on manifolds.Nevertheless, we found the isoperimetric stability of the boundary barycenter interesting in itself, as it contributes to a sharp understanding of the well-balancedness of almost minimizers of the isoperimetric inequality.Furthermore it shares some features with the strong form of isoperimetric stability recently developed by Fusco and Julin [6].Here is the bound we needed in [3], it is the main result of this note: Theorem 1 There exists a constant c ą 0 such that for any domain D with dpDq ď A{π, we have As observed by the referee, this estimate is clearly far from being optimal, since the l.h.s.can be zero with the r.h.s.being arbitrarily large.
Due to the invariance by translations and homotheties of the bound of Theorem 1, it is sufficient to show it when ρ " 1 and bpDq " 0.More precisely, translating by ´bpDq and applying the homothety of ratio a π{A, the above bound is equivalent to for any domain D with dpDq ď 1 and whose barycenter is 0. Due to Propositions 3 and 4 below, we are wondering if the exponent 1/4 in (4) could not replaced by 1{2 (or equivalently, replace A 1{4 d 1{4 pDq by a dpDq in Theorem 1).It would suffice to improve Lemma 9 below accordingly to obtain this conjecture.
We have not been very precise about the regularity assumption on the domain D, it should be such that the Bonnesen inequality [1] holds, as it is presented e.g. in the book of Burago and Zalgaller [2].In particular, the above result is true if the boundary C of the open set D is piecewise C 1 .Probably it can be extended to the framework of sets of finite perimeter, as defined in the lectures of Fusco [5].Then one has to be more careful with the definition of the boundary barycenter in (3): C has to be replaced by the reduced boundary B ˚D and σ by the total variation measure of the distributional derivative of the indicator function of D, see Fusco [5].
It could be tempting to extend Theorem 1 to the Euclidean spaces R n of dimension n ě 3.This is not possible, since the result is then wrong, as shown by the following example: where x 0 P p0, 1q, f : rx 0 , `8q Ñ R `is a decreasing function with f px 0 q " a 1 ´x2 0 andf pxq ą ?
1 ´x2 for all x ą x 0 .Here are the contributions of F to: ‚ the volume of D: π ş `8 x 0 f 2 puq du ‚ the area surface of D: 2π ş `8 x 0 f puq du ‚ the (unnormalized) barycenter of D: ´π ş `8 x 0 uf 2 puq du ¯p1, 0, 0q t ‚ the (unnormalized) barycenter of BD: ´2π ş `8 x 0 uf puq du ¯p1, 0, 0q t Let be given α ą 0 and consider the function g: @ u ą 0, gpuq u ´α For v ą 1, consider as function f the function g shifted by v: x 0 ą 0 is the solution of x 2 0 `g2 pv x0 q " 1 and for any u ě x 0 , we take f puq gpv `uq.for any α P p1, 2s, we get a counter-example to Theorem 1 by letting v go to `8.Similar considerations with α P p1{2, 1s enable to see why the Bonnesen inequality [1], recalled below in Theorem 5, is no longer valid in R 3 .It is replaced by an upper bound on the Fraenkel asymmetry index in Fusco, Maggi and Pratelli [4].The above construction also highlights the necessity of a restrictive assumption in Proposition 3 below.
These observations can easily be extended to the Euclidean spaces R n of dimension n ě 3. To avoid the pathologies of the previous example, one may want to work in the framework of compact Riemannian manifolds of dimension n ě 2. Then consider the subsets D with a fixed volume and a fixed renormalized Fréchet mean bpDq (replacing the notion of barycenter, in general bpDq will not be unique and one may have to consider their whole set).Assume that among such D, there is a minimizer B for the pn ´1q-Hausdorff measure of the boundary.There is no reason in general for the renormalized Fréchet mean bpBBq to coincide with bpBq.But, under bounds on the total diameter and on the curvature, one could try to evaluate the difference between bpBDq and bpBBq in terms of the isoperimetric deficit of D. This investigation is clearly out of the scope of the present note.
Nevertheless, in the restricted framework of nearly spherical sets, there is an extension (even an improvement) of Theorem 1 to Euclidean spaces of dimension n ě 2.An open set D from R n is said to be standard if its volume is equal to the volume of the unit ball B and if its barycenter bpDq is equal to 0. The standard set D is said to be nearly spherical if there exists a mapping u on the unitary sphere S BB centered at 0 such that

C
BD " tp1 `upxqqx : x P Su Define the barycenter of C as in (3): where σ is the pn ´1q-dimensional Hausdorff measure.The modified isoperimetric deficit is the non-negative quantity given by r dpDq σpCq ´σpSq When n " 2, this quantity is similar to the isoperimetric deficit dpDq defined in (2), at least when D is standard with dpDq P r0, 1s, in which case we have Indeed, we have, in one hand, and on the other hand, The interest of the (modified) isoperimetric deficit is: Proposition 3 There exist two constants pnq ą 0 and cpnq ą 0 depending only on n, such that for any standard nearly spherical set D with }u} W 1,8 pSq ď pnq, we have

Proof
This is an immediate consequence of Theorem 3.1 from Fusco [5], which finds two constants 1 pnq ą 0 and c 1 pnq ą 0 depending only on n, such that for any standard nearly spherical set D with }u} W 1,8 pSq ď 1 pnq, we have Proof From Lemma 3.3 from Fusco [5], we deduce that there exists a constant δpnq ą 0 such that any standard convex set D from R n with r dpDq ď δpnq is nearly spherical with }u} W 1,8 pSq ď pnq.Proposition 3 then shows that it is sufficient to take Cpnq cpnq to insure the validity of the above statement.

Proof of Theorem 1
In all this section, the set D will be as in the beginning of the introduction.
The arguments will be based on two results of the literature.The first one is quite old and is due to Bonnesen [1] (see also Theorem 1.3.1 of Burago and Zalgaller [2]): Theorem 5 Let r and R be the radii of the incircle and the circumcircle of D. We have π 2 pR ´rq 2 ď dpDq This result is not sufficient to deduce Theorem 1, since one can construct a set D whose boundary is included into the centered annulus of radii 1 ´ and 1 ` , with small ą 0, with a lot of folds in one direction, so that bpCq drifts in this direction, without bpDq moving a lot.
Thus we need a second result, due quite recently to Fusco and Julin [6].Let us recall their oscillation index βpDq, while referring to their paper for its motivation.To simplify the notation, assume that ρ " 1, i.e.A " π.Consider βpDq min where ν C pxq is the exterior unitary normal of C at x, under our assumption it is defined σ-a.s. on C (Fusco and Julin [6] defined it more generally for the sets of finite parameter, with the caution recalled after the statement of Theorem 1).Fusco and Julin [6] obtained the (multi-dimensional version of the) following result Theorem 6 Under the assumption A " π, there exists a constant r γ ą 0 such that Recalling the upper bound of (5) (which does note require dpDq ď 1), we deduce that if A " π, βpDq ď γ a dpDq (7) with γ r γ{ ?2π.With these ingredients at hand, we now come to the proof of Theorem 1.As already mentioned, it is sufficient to consider a standard set D with dpDq ď 1, for which the wanted bound reduces to (4) with a universal constant c ą 0.
Let us denote by o and O the respective centers of the incircle and the circumcircle of D. We begin by showing that o, O and 0 are quite close when the isoperimetric deficit is small.Lemma 7 As soon as D is a standard set with dpDq ď 1, we have maxt}o} , }O} }O ´o}u ă 3 a dpDq

Proof
Consider two numbers 0 ă r 1 ă R 1 and two points o 1 , O 1 P R 2 .If we want the inclusion of Bpo 1 , r 1 q into BpO 1 , R 1 q, we must have }O 1 ´o1 } ď R 1 ´r1 .Indeed, the equality in the previous bound (which is also its worse case) corresponds to the situation where Bpo 1 , r 1 q and BpO 1 , R 1 q are tangential at a point p which is at the intersection of Bpo 1 , r 1 q with BpO 1 , R 1 q.Then the three points p, O 1 and o 1 are on the same line and we have r `}O 1 ´o1 } `R " 2R, namely }O 1 ´o1 } " R 1 ´r1 .Since Bpo, rq Ă D Ă BpO, Rq, we deduce that }O ´o} ď R ´r ď a dpDq{π, according to Theorem 5.
Since the barycenter of D is 0, we have 0 " x dx ´żBpO,RqzD x dx It follows that We deduce that Due to the assumption dpDq ď 1 and from the fact that R ě 1, we have pπR 2 ´2R a dpDqq ě pπ ´2qR 2 , so that finally The triangle inequality enables to conclude to the last inequality: Our next step consists in checking that M, the set of minimizers in (6), is also close to 0. It was remarked by Fusco and Julin [6], as a simple consequence of the divergence theorem, that such minimizers coincide with the points y P R 2 maximizing the mapping It leads us to study the function f defined by where B is the unit disk centered at 0 and e 1 is the usual horizontal unit vector.
Lemma 8 The mapping f is decreasing and as t goes to 0 `,

Proof
For any t ě 0, we have with for any x 2 P r0, 1s, Differentiating with respect to t ě 0, for fixed x 2 P p0, 1q, we get The last expression is bounded uniformly in x 2 P r0, 1s and for t in a compact of R `zt1u.It follows that we can differentiate under the integral to get that for t ě 0, t " 1, This is sufficient to insure that f is decreasing on R `. Furthermore the above computation shows that uniformly over x 2 P r0, 1s, we have as t goes to 0 `, This implies that as t goes to 0 `, and next the last assertion of the lemma.
Note that by homothety and rotation, we have for any ą 0 and y P R 2 , ż Bp0, q In conjunction with the previous lemma, we deduce the following upper bound on the elements from M: Lemma 9 There exists a constant c ą 0 such that for any standard set D with dpDq ď 1, we have

Proof
It is sufficient to show that there exists P p0, 1s such that for any standard set D satisfying dpDq ď , we have where U D was defined in (8).Note that

Example 2
Consider the case n " 3 and the set D " B Y F , with B the unit open ball centered at 0 and F tpx, y, zq P R 3 : x ě x 0 and a y 2 `z2 ă f pxqu

CJacrψspyq " 1 S upyq} σpdyq ď c 3 ď c 1 pnqc 3 pnq a σpSq b r dpDq to get the announced result with cpnq c 1 pnqc 3 pnq{ a σpSq. The situation of convex sets is even simpler: Proposition 4
2 pSq ď c 1 pnq b r dpDqUp to replacing 1 pnq by pnq p1{2q ^ 1 pnq, we can assume that the mapping ψ : S Q y Þ Ñ p1 `upyqqy P BD is one-to-one.It enables to use the change of variable formula to get ż for the Jacobian of ψ at y P S. From the form of ψ, we deduce there exists a constant c 2 pnq ą 0, a function w : S Ñ R and a vector field v on S such that @ y P S, `wpyqupyq `xv, ∇ S uy pyq |wpyq| ď c 2 pnq }u} n´1 W 1,8 pSq }vpyq} ď c 2 pnq }u} n´1 W 1,8 pSq It follows that there exists a constant c 2 pnq ą 0 depending only on n such that as soon as }u} W 1,8 pSq ď pnq, we have @ y P S, }y ´ψpyqJacrψspyq} ď c 3 pnqp|upyq| `}∇ S upyq}q Thus we get that › pnq a σpSq }u} W 1,2 pSq where Cauchy-Schwarz' inequality was used in the last bound.It remains to write that }bpCq} " There exist two constants δpnq ą 0 and Cpnq ą 0 depending only on n, such that any standard convex set D from R n with r dpDq ď δpnq satisfies }bpCq} ď Cpnq b r dpDq Next let us find an upper bound on U D pyq, for y P R 2 not too small.We have 1s has been chosen above.An elementary computation shows that this is true with c 2 2p1 `πqp3 `2{πq.The end of the proof of Theorem 1 is immediate.Remark that by an application of the divergence theorem, we have ş C ν C pxq dx " 0, so that for any standard set D,Concerning the last term, use Theorem 5 and Lemma 7 to see that for x P C, if dpDq ď 1, on one hand, where (9) was taken into account.Assume that for some constant c 1 ą 0, }y} ě pc 1 `3qd 1{4 pDq, so that we are insured of}y} ě c 1 d 1{4 pDq `3d 1{2 pDq ě c 1 d 1{4 pDq `}O}C }x ´νC pxq} σpdxq Consider y P M and write }ν C pxq ´x} ď