The isoperimetric problem II

We introduce some methods to study the isoperimetric problem in 3-dimensional Riemannian manifolds and we show that in the positive curva- ture case we can control the topology of the isoperimetric regions. We consider the case of the projective space, which was first solved by Ritorea nd Ros, and we apply it to the Willmore problem. The isoperimetric problem is a classical topic in geometry but at the same time many basic questions about it remain unsolved. In this paper we first introduce some of the methods used in the study of that problem, although we will not try to be exhaustive at all. We will explain some relatively flexible ideas, like symmetrization or stability, which can be adapted to a certain number of situations. As an example we will study the problem for radial metrics on the 3-sphere. Then we will show that in 3-manifolds with positive Ricci curvature the topol- ogy of the isoperimetric regions can be controled. In particular we will prove that, when the volume of the ambient space is large, any isoperimetric surface must be either an sphere or a torus. As consequence, we will solve the isoperimetric problem in the real projective space or, equivalently, the isoperimetric problem for antipodal invariant regions in the 3-sphere. That result was first obtained by Ritorea nd Ros (35), but here we will give a somewhat different proof. Finally, as application of the above results, we will solve the Willmore conjec- ture for tori in euclidean space which are symmetric with respect to a point. 2. The isoperimetric problem In this paper we will only consider the three dimensional case. Let M be a Riemaniann 3-dimensional manifold with or without boundary and volume V(M ) ∈ )0, ∞). Given a positive number v< V(M ), we want to study the compact surfaces Σ ⊂ M such that

Schneider introduced an inter-dimensional difference body operator on convex bodies, and proved an associated inequality. In the prequel to this work, we showed that this concept can be extended to a rich class of operators from convex geometry and proved the associated isoperimetric inequalities. The role of cosine-like operators, which generate convex bodies in Rn from those in Rn, were replaced by inter-dimensional simplicial operators, which generate convex bodies in Rnm from those in Rn (or vice versa). In this work, we treat the Lp extensions of these operators, and, furthermore, extend the role of the simplex to arbitrary m-dimensional convex bodies containing the origin. We establish mth-order Lp isoperimetric inequalities, including the mth-order versions of the Lp Petty projection inequality, Lp Busemann-Petty centroid inequality, Lp Santaló inequalities, and Lp affine Sobolev inequalities. As an application, we obtain isoperimetric inequalities for the volume of the operator norm of linear functionals (Rn,‖⋅‖E)→(Rm,‖⋅‖F).

Various properties of isoperimetric, functional, Transport-Entropy and concentration inequalities are studied on a Riemannian manifold equipped with a measure, whose generalized Ricci curvature is bounded from below. First, stability of these inequalities with respect to perturbation of the measure is obtained. The extent of the perturbation is measured using several different distances between perturbed and original measure, such as a one-sided L ∞ bound on the ratio between their densities, Wasserstein distances, and Kullback–Leibler divergence. In particular, an extension of the Holley–Stroock perturbation lemma for the log-Sobolev inequality is obtained, and the dependence on the perturbation parameter is improved from linear to logarithmic. Second, the equivalence of Transport-Entropy inequalities with different cost-functions is verified, by obtaining a reverse Jensen type inequality. The main tool used is a previous precise result on the equivalence between concentration and isoperimetric inequalities in the described setting. Of independent interest is a new dimension independent characterization of Transport-Entropy inequalities with respect to the 1-Wasserstein distance, which does not assume any curvature lower bound.

By using optimal mass transport theory we prove a sharp isoperimetric inequality in $${\textsf {CD}} (0,N)$$ metric measure spaces assuming an asymptotic volume growth at infinity. Our result extends recently proven isoperimetric inequalities for normed spaces and Riemannian manifolds to a nonsmooth framework. In the case of n-dimensional Riemannian manifolds with nonnegative Ricci curvature, we outline an alternative proof of the rigidity result of Brendle (Comm Pure Appl Math 2021:13717, 2021). As applications of the isoperimetric inequality, we establish Sobolev and Rayleigh-Faber-Krahn inequalities with explicit sharp constants in Riemannian manifolds with nonnegative Ricci curvature; here we use appropriate symmetrization techniques and optimal volume non-collapsing properties. The equality cases in the latter inequalities are also characterized by stating that sufficiently smooth, nonzero extremal functions exist if and only if the Riemannian manifold is isometric to the Euclidean space.

This paper presents the proof of several inequalities by using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First, the author gives a new and simple proof of a lower bound of Berestycki, Nirenberg, and Varadhan concerning the principal eigenvalue of an elliptic operator with bounded measurable coefficients. The rest of the paper is a survey on the proofs of several isoperimetric and Sobolev inequalities using the ABP technique. This includes new proofs of the classical isoperimetric inequality, the Wulff isoperimetric inequality, and the Lions-Pacella isoperimetric inequality in convex cones. For this last inequality, the new proof was recently found by the author, Xavier Ros-Oton, and Joaquim Serra in a work where new Sobolev inequalities with weights came up by studying an open question raised by Haim Brezis.

Sobolev and isoperimetric inequalities with monomial weights

It is well known that isoperimetric inequalities imply in a very general measure-metric-space setting appropriate concentration inequalities. The former bound the boundary measure of sets as a function of their measure, whereas the latter bound the measure of sets separated from sets having half the total measure, as a function of their mutual distance. We show that under a lower bound condition on the Bakry--\'Emery curvature tensor of a Riemannian manifold equipped with a density, completely general concentration inequalities imply back their isoperimetric counterparts, up to dimension \emph{independent} bounds. As a corollary, we can recover and extend all previously known (dimension dependent) results by generalizing an isoperimetric inequality of Bobkov, and provide a new proof that under natural convexity assumptions, arbitrarily weak concentration implies a dimension independent linear isoperimetric inequality. Further applications will be described in a subsequent work. Contrary to previous attempts in this direction, our method is entirely geometric, continuing the approach set forth by Gromov and adapted to the manifold-with-density setting by Morgan.

In two-dimensional figure, the isoperimetric problem refers to finding two-dimensional figure that will produce the largest area among several shapes with equal perimeter. This research extends the isoperimetric problem to finding three-dimensional shapes with maximum volume among those having equal surface area. Our main goal is to solve the isoperimetric problem for prisms with regular n-sided base, prisms with irregular n-sided base and cylinder. In this research, the discussion is limited to prisms with regular and irregular bases. Our problem is equivalent with the problem of finding the smallest surface area of a given three-dimensional figure with the same volume. We will use a geometric approachin our proof. we will see the relationship between isoperimetric problems in two dimensional figures and isoperimetric problems in three-dimensional figure. We obtain the results of the isoperimetric problem from two prisms with regular n-sided bases and a prism with regular m-sided bases with n≤m, two prisms with regular n-sided bases and a prism with circular bases (cylinder), and two prisms with regular n-sided bases and a prism with irregular n-sided bases.

We investigate the logarithmic convexity of the length of the level curves for harmonic functions on surfaces and related isoperimetric type inequalities. The results deal with smooth surfaces, as well as with singular Alexandrov surfaces (also called surfaces with bounded integral curvature), a class which includes for instance surfaces with conical singularities and surfaces of CAT(0) type. Moreover, we study the geodesic curvature of the level curves and of the steepest descent for harmonic functions on surfaces with non-necessarily constant Gaussian curvature K. Such geodesic curvature functions turn out to satisfy certain Laplace-type equations and inequalities, from which we infer various maximum and minimum principles. The results are complemented by a number of growth estimates for the derivatives L' and L'' of the length of the level curve function L, as well as by examples illustrating the presentation. Our work generalizes some results due to Alessandrini, Longinetti, Talenti, Ma–Zhang and Wang–Wang.

After Hormander's fundamental paper on hypoellipticity [54], the study of partial differential equations arising from families of noncommuting vector fields has developed significantly. In this paper we study some basic functional and geometric properties of general families of vector fields that include the Hormander type as a special case. Similar to their classical counterparts, such properties play an important role in the analysis of the relevant differential operators (both linear and nonlinear). To motivate our results, we recall some classical inequalities. Let E C R be a Caccioppoli set (a measurable set having a locally finite perimeter); then one has the isoperimetric inequality

Affine vs. Euclidean isoperimetric inequalities

We establish lower estimates for an integral functional$$\int\limits_\Omega f(u(x), \nabla u(x)) \, dx ,$$where \(\Omega\) -- a bounded domain in \(\mathbb{R}^n \; (n \geqslant 2)\), an integrand \(f(t,p) \, (t \in [0, \infty),\; p \in \mathbb{R}^n)\) -- a function that is \(B\)-measurable with respect to a variable \(t\) and is convex and even in the variable \(p\), \(\nabla u(x)\) -- a gradient (in the sense of Sobolev) of the function \(u \colon \Omega \rightarrow \mathbb{R}\). In the first and the second sections we utilize properties of permutations of differentiable functions and an isoperimetric inequality \(H^{n-1}( \partial A) \geqslant \lambda(m_n A)\), that connects \((n-1)\)-dimensional Hausdorff measure \(H^{n-1}(\partial A )\) of relative boundary \(\partial A\) of the set \(A \subset \Omega\) with its \(n\)-dimensional Lebesgue measure \(m_n A\). The integrand \(f\) is assumed to be isotropic, i.e. \(f(t,p) = f(t,q)\) if \(|p| = |q|\).Applications of the established results to multidimensional variational problems are outlined. For functions \( u \) that vanish on the boundary of the domain \(\Omega\), the assumption of the isotropy of the integrand \( f \) can be omitted. In this case, an important role is played by the Steiner and Schwartz symmetrization operations of the integrand \( f \) and of the function \( u \). The corresponding variants of the lower estimates are discussed in the third section. What is fundamentally new here is that the symmetrization operation is applied not only to the function \(u\), but also to the integrand \(f\). The geometric basis of the results of the third section is the Brunn-Minkowski inequality, as well as the symmetrization properties of the algebraic sum of sets.

On the structure of subsets of the discrete cube with small edge boundary, Discrete Analysis 2018:9, 29 pp. An isoperimetric inequality is a statement that tells us how small the boundary of a set can be given the of the set, for suitable notions of size and boundary. For example, one formulation of the classical isoperimetric inequality in $\mathbb R^n$ is as follows. Given a subset $X$ of $\mathbb R^n$, define the $\epsilon$-_expansion_ of $X$ to be the open set $X_\epsilon=\{y\in\mathbb R^n: d(y,X)<\epsilon\}$. If in addition $X$ is measurable, define the of its boundary to be $\lim\inf_{\epsilon\to 0} \epsilon^{-1}\mu(X_\epsilon\setminus X)$. (If $X$ is a set with a suitably smooth topological boundary $\partial X$, then this turns out to equal the surface measure of $\partial X$.) Then amongst all sets $X$ of a given measure, the one with the smallest boundary is an $n$-dimensional ball. Isoperimetric inequalities have been the focus of a great deal of research, partly for their intrinisic interest, but also because they have numerous applications. One particularly useful one is the _edge-isoperimetric inequality in the discrete cube_. This concerns subsets $X$ of the $n$-dimensional cube $\{0,1\}^n$, which we turn into a graph by joining two points $x$ and $y$ if they differ in exactly one coordinate. The of a set $X$ is simply its cardinality, the _edge-boundary_ of $X$ is defined to be the set of edges between $X$ and its complement, and the of the edge-boundary is the number of such edges. If $|X|=2^d$, then it is known that the edge-boundary is minimized when $X$ is a $d$-dimensional subspace of $\mathbb F_2^n$ generated by $d$ standard basis vectors. More generally, if $|X|=m$, then the edge-boundary is minimized when $X$ is an initial segment in the lexicographical ordering, which is the ordering where we set $x<y$ if $x_i<y_i$ for the first coordinate $i$ where $x_i$ and $y_i$ differ. (This coincides with the ordering we obtain if we think of the sequences as binary representations of integers.) For the two isoperimetric inequalities just mentioned, as well as many others, it is known that the extremal examples provided are essentially the only ones: a subset of $\mathbb R^n$ with a boundary that is as small as possible has to be an $n$-dimensional ball, and a subset of the discrete cube with edge-boundary that is as small as possible has to be an initial segment of the lexicographical ordering, up to the symmetries of the graph. Furthermore, there are _stablity_ results: a set with a boundary that is _almost_ as small as possible must be close to an extremal example. Such a result tells us that the isoperimetric inequalities are robust, in the sense that if you slightly perturb the condition on the set, then you only slightly perturb what the set has to look like. This paper is about an extremely precise stability result for the edge isoperimetric inequality in the discrete cube. There have been a number of papers on such results (see the introduction to the paper for details), but they have been mainly for sets of $2^d$ for some $d$, where the goal is to prove that they must be close to $d$-dimensional subcubes -- that is, subspaces (or their translates) generated by $d$ standard basis vectors. This paper considers sets of arbitrary and proves the following result. Suppose that $X$ is a subset of $\{0,1\}^n$ of $m$. Suppose that the of the edge-boundary of $X$ is at most $g_n(m)+l$, where $g_n(m)$ is the of the edge-boundary of the initial segment $I_m$ of $m$ in the lexicographical order. Then there is an automorphism $\phi$ of $\{0,1\}^n$ (meaning a bijection that takes neighbouring points to neighbouring points) such that $|X\Delta\phi(I_m)|\leq Cl$, where $C$ is an absolute constant. They give an example to show that $C$ must be at least 2, and thus that their result is best possible up to the value of the constant $C$. Previous proofs of stability versions of the edge-isoperimetric inequality in the cube have used Fourier analysis. The proof in this paper uses purely combinatorial methods, such as induction on the dimension, and compressions. To get these methods to work, several interesting ideas are needed, including some new results about the influence of variables.

Discrete isoperimetric variational problems which model single and double integral isoperimetric problems are formulated and some multiplier rules are derived. For quardratic functionals, the Euler-Lagrange equation is linear and can be analyzed bydeterminant methods. difference equations methods. or numerically bythealgebraic eigenvalue problem. A specific example is given wherethe eigenfuntions and eigenvalue of this discrete problem converge. as the step size goes to zero, to the eigenfunctionsand eigenvalues of the corresponding, continuous problem Recent developments in algebraic geometry offer the hopeof using Groebner (=Gröbner or Grobner) basis methods to numerically solve some systems of equationsgenerated by Lagrange multiplier methods. These methods mayapply to nonlinear systems of equations whenever they can be reduced to systems of polynomial equations. The Groebner basis algorithm of Buchberger is an extension of Gaussian elimination to polynomial systems.

Isoperimetric problems are of importance in engineering applications, where it is often desirable to maximize or minimize some physical variable by shape variation, subject to geometrical constraints, such as keeping an area or volume constant. The calculus of variations can offer a powerful tool for the solution of such problems where there is a governing variational minimum or maximum principle, e.g. Helmholtz's principle in slow viscous flow. In these problems the well-known Euler equations derived by the calculus of variations are supplemented with additional boundary conditions arising from the shape variation, as well as the usual physical boundary conditions. The exact solution of such unknown boundary problems can be difficult to find. A good approach then is to apply a complementary extremum principle that offers an algorithm for determining bounds on the exact extremal value of the original functional. This paper shows how this may be done in the case of the fundamental problem of the calculus of variations with variable endpoints. We apply this approach to a simple engineering problem of a stretched spring.