Bodies In Rn Research Articles

Higher-order Lp isoperimetric and Sobolev inequalities

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).

On bodies in [formula omitted] with congruent sections by cones or non-central planes

Let K and L be two convex bodies in Rn, n≥3 such that their sections by cones {x∈Rn:x⋅ξ=t|x|} or non-central planes with a fixed distance from the origin are directly congruent. We prove that if their boundaries are of class C2, then K and L coincide.

Optimal recovery and volume estimates

We study volumes of sections of convex origin-symmetric bodies in Rn induced by orthonormal systems on probability spaces. The approach is based on volume estimates of John-Löwner ellipsoids and expectations of norms induced by the respective systems. The estimates obtained allow us to establish lower bounds for the radii of sections which gives lower bounds for Gelfand widths (or linear cowidths). As an application we offer a new method of evaluation of Gelfand and Kolmogorov widths of multiplier operators. In particular, we establish sharp orders of widths of standard Sobolev classes Wpγ, γ>0 in Lq on two-point homogeneous spaces in the difficult case, i.e. if 1<q≤p≤∞.

Extremizers in Soprunov and Zvavitch's Bezout inequalities for mixed volumes

In [20], I. Soprunov and A. Zvavitch have translated the Bezout inequalities (from Algebraic Geometry) into inequalities of mixed volumes satisfied by the simplex. They conjecture this set of inequalities characterizes the simplex, among all convex bodies in Rn. Together with C. Saroglou, they proved the characterization among all polytopes [17] and, for a larger set of inequalities, among all convex bodies [18]. The conjecture remains open for n≥4. In this work, we investigate necessary conditions on the structure of the boundary of a convex body K, for K to satisfy all inequalities. In particular, we obtain a new solution of the 3-dimensional case.

An intrinsic volume metric for the class of convex bodies in ℝn

A new intrinsic volume metric is introduced for the class of convex bodies in [Formula: see text]. As an application, an inequality is proved for the asymptotic best approximation of the Euclidean unit ball by arbitrarily positioned polytopes with a restricted number of vertices under this metric. This result improves the best known estimate, and shows that dropping the restriction that the polytope is contained in the ball or vice versa improves the estimate by at least a factor of dimension. The same phenomenon has already been observed in the special cases of volume, surface area and mean width approximation of the ball.

TheLpchord Minkowski problem

AbstractChord measures are newly discovered translation-invariant geometric measures of convex bodies inRn{{\mathbb{R}}}^{n}, in addition to Aleksandrov-Fenchel-Jessen’s area measures. They are constructed from chord integrals of convex bodies and random lines. Prescribing theLp{L}_{p}chord measures is called theLp{L}_{p}chord Minkowski problem in theLp{L}_{p}Brunn-Minkowski theory, which includes theLp{L}_{p}Minkowski problem as a special case. This article solves theLp{L}_{p}chord Minkowski problem whenp&gt;1p\gt 1and the symmetric case of0&lt;p&lt;10\lt p\lt 1.

Lp-moment mixed quermassintegrals

Lp-moment mixed quermassintegrals of convex bodies in Rn are introduced. The Brunn-Minkowski type inequality and Aleksandrov-Fenchel type inequality are established for Lp-moment mixed quermassintegrals that imply affine mixed quermassintegrals inequality, Lutwak?s mixed polar projection inequality, and isoperimetric inequality for Lp-moment mixed quermassintegrals. Inequalities of Lp-moment mixed quermassintegrals of polar bodies are proved.

On the continuity of the solution to the Minkowski problem for Lp torsional measure

This paper deals with on the continuity of the solution to the Minkowski problem for Lp torsional measure. For p ? (1, n + 2) ? (n + 2,?), we show that a sequence of convex bodies in Rn is convergent in Hausdorff metric if the sequence of the Lp torsional measures (associated with these convex bodies) is weakly convergent. Moreover, we also prove that the solution to the Minkowski problem for Lp torsional measure is continuous with respect to p.

On the stability of the Lp-curvature

It is known that the Lp-curvature of a smooth, strictly convex body in Rn is constant only for origin-centered balls when 1≠p>−n, and only for balls when p=1. If p=−n, then the L−n-curvature is constant only for origin-symmetric ellipsoids. We prove ‘local’ and ‘global’ stability versions of these results. For p≥1, we prove a global stability result: if the Lp-curvature is almost a constant, then the volume symmetric difference of K˜ and a translate of the unit ball B is almost zero. Here K˜ is the dilation of K with the same volume as the unit ball. For 0≤p<1, we prove a similar result in the class of origin-symmetric bodies in the L2-distance. In addition, for −n<p<0, we prove a local stability result: There is a neighborhood of the unit ball that any smooth, strictly convex body in this neighborhood with ‘almost’ constant Lp-curvature is ‘almost’ the unit ball. For p=−n, we prove a global stability result in R2 and a local stability result for n>2 in the Banach-Mazur distance.

Local 𝐿^{𝑝}-Brunn–Minkowski inequalities for 𝑝&lt;1

The L p L^p -Brunn–Minkowski theory for p ≥ 1 p\geq 1 , proposed by Firey and developed by Lutwak in the 90’s, replaces the Minkowski addition of convex sets by its L p L^p counterpart, in which the support functions are added in L p L^p -norm. Recently, Böröczky, Lutwak, Yang and Zhang have proposed to extend this theory further to encompass the range p ∈ [ 0 , 1 ) p \in [0,1) . In particular, they conjectured an L p L^p -Brunn–Minkowski inequality for origin-symmetric convex bodies in that range, which constitutes a strengthening of the classical Brunn-Minkowski inequality. Our main result confirms this conjecture locally for all (smooth) origin-symmetric convex bodies in R n \mathbb {R}^n and p ∈ [ 1 − c n 3 / 2 , 1 ) p \in [1 - \frac {c}{n^{3/2}},1) . In addition, we confirm the local log-Brunn–Minkowski conjecture (the case p = 0 p=0 ) for small-enough C 2 C^2 -perturbations of the unit-ball of ℓ q n \ell _q^n for q ≥ 2 q \geq 2 , when the dimension n n is sufficiently large, as well as for the cube, which we show is the conjectural extremal case. For unit-balls of ℓ q n \ell _q^n with q ∈ [ 1 , 2 ) q \in [1,2) , we confirm an analogous result for p = c ∈ ( 0 , 1 ) p=c \in (0,1) , a universal constant. It turns out that the local version of these conjectures is equivalent to a minimization problem for a spectral-gap parameter associated with a certain differential operator, introduced by Hilbert (under different normalization) in his proof of the Brunn–Minkowski inequality. As applications, we obtain local uniqueness results in the even L p L^p -Minkowski problem, as well as improved stability estimates in the Brunn–Minkowski and anisotropic isoperimetric inequalities.

The measure-comparison problem for polar (p,μ)-centroid bodies

For p>−1 and α-homogeneous measure μ on Rn with α≠−p, this paper considers the measure-comparison problem for the polar (p,μ)-centroid operator Γp,μ⁎. Namely, if K,L are origin-symmetric convex bodies in Rn with satisfyingΓp,μ⁎K⊂Γp,μ⁎L, does it follow thatpα+pμ(L)p≤pα+pμ(K)p? This paper shows that the answer is confirmative when −1<p<0,α+p<0, and negative when p≥0, or −1<p<0,α+p>0. In the negative cases, we also give a solution to the isomorphic version of the above problem. Namely, the condition Γp,μ⁎K⊂Γp,μ⁎L implies thatμ(L)1/α≤nμ(K)1/α.

A Riesz representation theorem for functionals on log-concave functions

The classic Riesz representation theorem characterizes all linear and increasing functionals on the space Cc(X) of continuous compactly supported functions. A geometric version of this result, which characterizes all linear increasing functionals on the set of convex bodies in Rn, was essentially known to Alexandrov. This was used by Alexandrov to prove the existence of mixed area measures in convex geometry.In this paper we characterize linear and increasing functionals on the class of log-concave functions on Rn. Here “linear” means linear with respect to the natural addition on log-concave functions which is the sup-convolution. Equivalently, we characterize pointwise-linear and increasing functionals on the class of convex functions. For some choices of the exact class of functions we prove that there are no non-trivial such functionals. For another choice we obtain the expected analogue of the result for convex bodies. And most interestingly, for yet another choice we find a new unexpected family of such functionals.Finally, we explain the connection between our results and recent work done in convex geometry regarding the surface area measure of a log-concave functions. An application of our results in this direction is also given.

Chord length distribution and the distance between two random points in a convex body in Rn

In this article for n-dimensional convex body D the relation between the chord length distribution function and the distribution function of the distance between two random points in D was found. Also the relation between their moments was found.

Asymptotic normality for random simplices and convex bodies in high dimensions

Central limit theorems for the log-volume of a class of random convex bodies in R n \mathbb {R}^n are obtained in the high-dimensional regime, that is, as n → ∞ n\to \infty . In particular, the case of random simplices pinned at the origin and simplices where all vertices are generated at random is investigated. The coordinates of the generating vectors are assumed to be independent and identically distributed with subexponential tails. In addition, asymptotic normality is also established for random convex bodies (including random simplices pinned at the origin) when the spanning vectors are distributed according to a radially symmetric probability measure on the n n -dimensional ℓ p \ell _p -ball. In particular, this includes the cone and the uniform probability measure.

The radii of sections of origin-symmetric convex bodies and their applications

Let V and W be any convex and origin-symmetric bodies in Rn . Assume that for some A ∈LRn→Rn, detA≠0, V is contained in the ellipsoid A−1B2n, where B2n is the unit Euclidean ball. We give a lower bound for the W-radius of sections of A−1V in terms of the spectral radius of A∗A and the expectations of ‖⋅‖V and ‖⋅‖Wo with respect to Haar measure on Sn−1⊂Rn. It is shown that the respective expectations are bounded as n→∞ in many important cases. As an application we offer a new method of evaluation of n-widths of multiplier operators. As an example we establish sharp orders of n-widths of multiplier operators Λ:LpMd→LqMd, 1<q≤2≤p<∞ on compact homogeneous Riemannian manifolds Md. Also, we apply these results to prove the existence of flat polynomials on Md.

Some Hermite–Hadamard type integral inequalities for convex functions defined on convex bodies in ℝn\mathbb{R}^{n}

Abstract In this paper, by the use of the divergence theorem, we establish some integral inequalities of Hermite–Hadamard type for convex functions of several variables defined on closed and bounded convex bodies in the Euclidean space ℝ n {\mathbb{R}^{n}} for any n ≥ 2 {n\geq 2} .

The dual Minkowski problem for symmetric convex bodies

The dual Minkowski problem for even data asks what are the necessary and sufficient conditions on a prescribed even measure on the unit sphere for it to be the q-th dual curvature measure of an origin-symmetric convex body in Rn. A full solution to this is given when 1<q<n. The necessary and sufficient conditions turn out to be an explicit measure concentration condition. To obtain the results, a variational approach is used, where the functional is the sum of a dual quermassintegral and an entropy integral. The proof requires two crucial estimates. The first is an estimate of the entropy integral which is obtained by using a spherical partition. The second is a sharp estimate of the dual quermassintegrals for a carefully chosen barrier convex body.

A sausage body is a unique solution for a reverse isoperimetric problem

We consider the class of λ-concave bodies in Rn+1; that is, convex bodies with the property that each of their boundary points supports a tangent ball of radius 1/λ that lies locally (around the boundary point) inside the body. In this class we solve a reverse isoperimetric problem: we show that the convex hull of two balls of radius 1/λ (a sausage body) is a unique volume minimizer among all λ-concave bodies of given surface area. This is in a surprising contrast to the standard isoperimetric problem for which, as it is well-known, the unique maximizer is a ball. We solve the reverse isoperimetric problem by proving a reverse quermassintegral inequality, the second main result of this paper.

Orlicz-Aleksandrov-Fenchel Inequality for Orlicz Multiple Mixed Volumes

Our main aim is to generalize the classical mixed volumeV(K1,…,Kn)and Aleksandrov-Fenchel inequality to the Orlicz space. In the framework of Orlicz-Brunn-Minkowski theory, we introduce a new affine geometric quantity by calculating the Orlicz first-order variation of the mixed volume and call itOrlicz multiple mixed volumeof convex bodiesK1,…,Kn, andLn, denoted byVφ(K1,…,Kn,Ln), which involves(n+1)convex bodies inRn. The fundamental notions and conclusions of the mixed volume and Aleksandrov-Fenchel inequality are extended to an Orlicz setting. The related concepts and inequalities ofLp-multiple mixed volumeVp(K1,…,Kn,Ln)are also derived. The Orlicz-Aleksandrov-Fenchel inequality in special cases yieldsLp-Aleksandrov-Fenchel inequality, Orlicz-Minkowski inequality, and Orlicz isoperimetric type inequalities. As application, a new Orlicz-Brunn-Minkowski inequality for Orlicz harmonic addition is established, which implies Orlicz-Brunn-Minkowski inequalities for the volumes and quermassintegrals.

Hyperspaces of smooth convex bodies up to congruence

We determine the homeomorphism type of the hyperspace of positively curved C∞ convex bodies in Rn, and derive various properties of its quotient by the group of Euclidean isometries. We make a systematic study of hyperspaces of convex bodies that are at least C1. We show how to destroy the symmetry of a family of convex bodies, and prove that this cannot be done modulo congruence.