A reverse isoperimetric inequality and extremal theorems 3 1. A euclidean conemetric g on a closed surface m is a path metric structure such that every point has a neighborhood isometric either to an open euclidean disk or to a neighborhood of the apex of a euclidean cone with angle. A natural issue arising from the optimality of the ball in the isoperimetric inequality, is that of stability estimates of the type pe e. Find out information about isoperimetric inequality. You might try using analysis of boolean functions whenever youre faced with a problems involving boolean strings in which both the uniform probability distribution and the hamming graph structure play a role. The description for this book, isoperimetric inequalities in mathematical physics. An elementary proof of the isoperimetric inequality. Among all planar regions with a given perimeter p, the circle encloses the greatest area. We stated the isoperimetric inequality without proof.
The rst proof, presented in chapter 1, closely follows the original one and uses. I dont think this can be done, but i thought id see if others have an idea. We will also show that, in a way, steiner symmetrization could be used as a useful tool to prove pettys conjectured projection inequality. Our main result states that we can study the cheeger isoperimetric inequality in a riemann surface by using a graph related to it, even if the surface has injectivity radius zero this graph is inspired in kanais. The curves c1t and c2t denote semicircles figure 2. If v is the volume of a closed, threedimensional region, and a is its surface area, then the following inequality always holds. The isoperimetric inequality 1 holds for any domain on a totally geodesic surface in m. Equality in 1 is attained for a nonregular object a domain isometric to the lateral surface of a right circular cone with complete angle about the vertex. A simple proof of an isoperimetric inequality for euclidean.
The isoperimetric inequality 1 is valid also for a twodimensional manifold of bounded curvature, which is a more general type of manifold than a riemannian manifold. The isoperimetric inequality states the intuitive fact that, among all shapes with a given surface area, a sphere has the maximum volume. Stephen demjanenko 1 introduction the isoperimetric problem can be stated two ways. The isoperimetric inequality on a surface springerlink.
In the absence of any restriction on shape, the curve is a circle. A reverse isoperimetric inequality, stability and extremal. The first part was proved independently by weil and by beckenbach and rado. This result, which is also known as the isoperimetric inequality, dates back to antiquity.
In the present paper we address the stability of the isoperimetric inequality on the sphere by schmidt 21 stating that if e. W e pro ve a new isoperimetric inequality which relates the area of a multiply connected curv ed surf ace, its euler characteristic, the length of its boundary, and its gaussian curv ature. We prove a bestpossible isoperimetric inequality for a. We prove a new isoperimetric inequality which relates the area of a multiply connected curved surface, its euler characteristic, the length of its boundary, and its gaussian curvature. Gaussian hypercontractivity theorem, gaussian isoperimetric inequality, gaussian noise operator, gaussian space, gaussian surface area, hermite polynomials.
There are two extreme ways of proving such an equality. We showed above that a1 a2, therefore it is enough to show wlog that c1t is a semicircle. First, it nicely explains the story of the classical isoperimetric inequality, a result with a big disproportion between the ease of formulation and difficulty of the proof. In mathematics, the isoperimetric inequality is a geometric inequality involving the surface area of a set and its volume. In dimensional space the inequality lower bounds the surface area or perimeter of a set. Changing the angle to maximise the area again, look at q1 and q2. In words, a set of size n minimizing the edge boundary is l dn. Consider a region ofccomponents, euler characteristic.
Snis a measurable set having the same measure as a. Isoperimetric problem, in mathematics, the determination of the shape of the closed plane curve having a given length and enclosing the maximum area. Annals of mathematics studies 27 book 27 paperback. Mar 28, 20 here is an application of the spherical isoperimetric inequality.
The classical isoperimetric inequality is as follows. In addition, the isoperimetric inequality, a blaschkesantalo type inequality, and the monotonicity inequality for the dual orlicz geominimal surface areas are established. The curveshortening flow decreases the isoperimetric ratio of any smooth convex curve so that, in the limit as the curve shrinks to a point, the ratio becomes 4 2 for higherdimensional bodies of dimension d, the isoperimetric ratio can similarly be defined as b d v d. R3 is a regular surface, if for each p 2s there is a neighborhood v of p in r3 and a map x. Let be the inner diameter and, where is a closed loop. Thus we know that a circle maximizes enclosed area among all smooth regular simple closed curves of the same length. The blog has been pretty quiet the last few weeks with the usual endofterm business, research, and aexams mine is coming up quite soon. A celebrated theorem of kanai states that quasiisometries preserve isoperimetric inequalities between uniform riemannian manifolds with positive injectivity radius and graphs. This talk explores a proof of this fact for subsets of rn via the brunnminkowski theorem. The purpose of this expository paper is to collect some mainly recent inequalities, conjectures, and open questions closely related to isoperimetric problems in real, finitedimensional banach spaces minkowski spaces.
May 16, 2008 the blog has been pretty quiet the last few weeks with the usual endofterm business, research, and aexams mine is coming up quite soon. On isoperimetric inequalities in minkowski spaces journal. To the joy of analysts everywhere, we can rephrase this theorem as an inequality. Most books on convexity also contain a discussion of the isoperimetric inequality from that perspective. Princeton university press august 21, 1951 language. Lets take a closed compact surface with a riemannian metric, which induces an inner metric. Isoperimetric inequality isoperimetric nequality is a wellknown statement in the following form. It is a dimensionless quantity that is invariant under similarity transformations of the curve according to the isoperimetric inequality, the isoperimetric ratio has its minimum value, 4. I was looking through some of my notes recently and came upon two very short fourier analysis proofs of the isoperimetric inequality. Two cute proofs of the isoperimetric inequality the. We present an elementary proof of the known inequality l2. The isoperimetric inequality on a surface recei ved.
A new isoperimetric inequality and the concentration of. Second, regarding the proof as a whole, it seems useful to think of it as a way of transforming the difficult global optimization problem implied by the isoperimetric inequality how to enclose the greatest possible area within a given circumference into a trivial local optimization problem through some clever bookkeeping. Curves with weakly bounded curvature let be 2manifold of class c2. Pdf isoperimetric inequalities for lp geominimal surface. One doesnt need to assume this much smoothness on the boundary for the isoperimetric inequality to hold, recti ability su ces b,g. Now let 0 and we get the isoperimetric inequality for the given curve c, l2 4. An isoperimetric inequality for antipodal subsets of the. We say a family of subsets of 1,2,n is antipodal if it is closed under taking complements. The book description for the forthcoming isoperimetric inequalities in mathematical physics. I am looking for a proof using the calculus of variations in the spirit of the proof of the standard isoperimetric inequality on the plane. In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume.
Bakry and ledoux gave another proof of bobkovs functional inequality based on the semigroup techniques which. In your first sketch, you are comparing with a large circle and saying that your actual area is smaller, while in the second sketch you are comparing. An elementary proof of the isoperimetric inequality nikolaos dergiades abstract. This second part contains deep results obtained by the author. Specifically, the isoperimetric inequality states, for the length l of a closed curve and the area a of the.
In analytic geometry, the isoperimetric ratio of a simple closed curve in the euclidean plane is the ratio l 2 a, where l is the length of the curve and a is its area. The statement that the area enclosed by a plane curve is equal to or less than the square of its perimeter divided by 4. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Fact you cant cut up a beach ball into equal parts with a path that is too short lets take a closed compact surface with a riemannian metric, which induces an inner metric.
Examples of how to use isoperimetric in a sentence from the cambridge dictionary labs. In mathematics, the isoperimetric inequality is a geometric inequality involving the square of the circumference of a closed curve in the plane and the area of a plane region it encloses, as well as its various generalizations. For every riemannian metric on a disk b 2 with k x. For every riemannian metric on a disk b 2 with the gauss curvature k x.
More generally, the tools may still apply when studying functions on or subsets of product probability spaces. Isoperimetric inequalities in mathematical physics gyorgy. In n dimensional space r n the inequality lower bounds the surface area or perimeter p e r of a set s. Isoperimetric inequality encyclopedia of mathematics. The loops in the infimum are bisecting curves which split the sphere into two regions of. In mathematics, the gaussian isoperimetric inequality, proved by boris tsirelson and vladimir sudakov, and later independently by christer borell, states that among all sets of given gaussian measure in the ndimensional euclidean space, halfspaces have the minimal gaussian boundary measure. Since the circle of the same length l bounds a region of area a 0 satisfying l 2 4 a 0, we know that a a 0. May 09, 20 case of the gaussian isoperimetric inequality 7, to the almgren higher codimension isoperimetric inequality 2, 5 and to several other isoperimetric problems 3, 1, 9. Pdf the isoperimetric inequality for minimal surfaces in a.
Isoperimetric inequality on the sphere via calculus of. Isoperimetric literally means having the same perimeter. Isoperimetric inequalities in mathematical physics. Curve surface gaussian curvature euler characteristic isoperimetric inequality these keywords were added by machine and not by the authors. A proof of the isoperimetric inequality how does it work. By the classical isoperimetric inequality in rn, pe is nonnegative and zero if and only if ecoincides with b e up to null sets and to a translation. Quantitative isoperimetric inequalities for anisotropic surface energies are proven where the isoperimetric deficit controls both the fraenkel asymmetry and a measure of the oscillation of the. This lower bound depends only on an upper bound for the absolute mean curvature function of m, an upper bound of the absolute sectional curvature of n and a lower bound for the injectivity radius of n. We give an elementary proof of the isoperimetric inequality for polygons, simplifying the proof given by t. An isoperimetric inequality for surfaces whose gaussian curvature is bounded above, siberian math. Among all bodies in in space in plane with a given volume given area, the one with the least surface area least perimeter is the ball the disk.
The calculus of variations evolved from attempts to solve this problem and the. The spherical isoperimetric inequality parker glynnadey. The quantitative isoperimetric inequality and related topics. The theorem has generalizations to higher dimensions, and. The calculus of variations evolved from attempts to solve this problem and the brachistochrone leasttime problem in 1638 the italian mathematician and astronomer. A history of the problem, proofs and applications april 29, 2008 by. A functional form of the isoperimetric inequality for the gaussian measure.
The equality holds only when is a ball in on a plane, i. Citeseerx the isoperimetric inequality on a surface. The theorem has generalizations to higher dimensions, and even has many variants in two dimensions. Let g be a domain with compact closure g on a complete possibly compact orientable riemannian surface m with a c 2 smooth metric. Here is an application of the spherical isoperimetric inequality.
In his book global methods for combinatorial isoperimetric problems, harper o ers two proofs of theorem 1. Pdf the isoperimetric inequality for minimal surfaces in. Isoperimetric inequality article about isoperimetric. A, where l and a are the perimeter and the area of a polygon. Isoperimetric inequality and area growth of surfaces with.
Using 1, inequalities can be established for the length of a. Among all regions in the plane, enclosed by a piecewise c1 boundary curve, with area a and perimeter l, 4. Fact you cant cut up a beach ball into equal parts with a path that is too short. If m has nonempty boundary, then we require that every boundary point has a. Since a totally geodesic surface is minimal in m, it has been naturally conjectured that 1 should. On account of a 27 it follows finally that these arcs are subarcs of the same great circle. In this paper we prove a quantitative version of the isoperimetric inequality on the sphere with a constant independent of the volume of the set e. Pdf isoperimetric inequalities for lp geominimal surface area. This process is experimental and the keywords may be updated as the learning algorithm improves. In this thesis, we give a lower bound on the areas of small geodesic balls in an immersed hypersurface m contained in a riemannian manifold n.