Abstract Let and let and be two convex bodies in such that their orthogonal projections and onto any ‐dimensional subspace are directly congruent, that is, there exists a rotation and a vector such that . Assume also that the 2‐dimensional projections of and are pairwise different and they do not have ‐symmetries. Then and are congruent. We also prove an analogous more general result about twice differentiable functions on the unit sphere in .

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

Optimal recovery and volume estimates

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

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.

Abstract Chord 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>1p\gt 1and the symmetric case of0<p<10\lt p\lt 1.

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.

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.

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.

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

A Riesz representation theorem for functionals on log-concave functions

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.

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

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

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

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