We study the metric projection onto the closed convex cone in a real Hilbert space $\mathscr {H}$ generated by a sequence $\mathcal {V} = \{v_n\}_{n=0}^\infty $. The first main result of this article provides a sufficient condition under which the closed convex cone generated by $\mathcal {V}$ coincides with the following set:

$$ \begin{align*} \mathcal{C}[[\mathcal{V}]]: = \bigg\{\sum_{n=0}^\infty a_n v_n\Big|a_n\geq 0,\text{ the series }\sum_{n=0}^\infty a_n v_n\text{ converges in } \mathscr{H}\bigg\}. \end{align*} $$

$\mathcal {C}[[\mathcal {V}]]$

$L^2([-1,1])$

A. R. Martínez Fernández obtained upper bounds for quermassintegrals of the p-inner parallel bodies: an extension of the classical inner parallel body to the $L_p$ -Brunn-Minkowski theory. In this paper, we establish (sharp) upper and lower bounds for quermassintegrals of p-inner parallel bodies. Moreover, the sufficient and necessary conditions of the equality case for the main inequality are obtained, which characterize the so-called tangential bodies.

In the present paper, we first introduce the concepts of the Lpq-capacity measure and Lp mixed q-capacity and then prove some geometric properties of Lpq-capacity measure and a Lp Minkowski inequality for the q-capacity for any fixed p ⩾ 1 and q > n. As an application of the Lp Minkowski inequality mentioned above, we establish a Hadamard variational formula for the q-capacity under p-sum for any fixed p ⩾ 1 and q > n, which extends results of Akman et al. (Adv. Calc. Var. (in press)). With the Hadamard variational formula, variational method and Lp Minkowski inequality mentioned above, we prove the existence and uniqueness of the solution for the Lp Minkowski problem for the q-capacity which extends some beautiful results of Jerison (1996, Acta Math.176, 1–47), Colesanti et al. (2015, Adv. Math.285, 1511–588), Akman et al. (Mem. Amer. Math. Soc. (in press)) and Akman et al. (Adv. Calc. Var. (in press)). It is worth mentioning that our proof of Hadamard variational formula is based on Lp Minkowski inequality rather than the direct argument which was adopted by Akman (Adv. Calc. Var. (in press)). Moreover, as a consequence of Lp Minkowski inequality for q-capacity, we get an interesting isoperimetric inequality for q-capacity.

A new approach to prove weak convergence of random polytopes on the space of compact convex sets is presented. This is used to show that the profile of the rescaled Schläfli random cone of a random conical tessellation, generated by n independent and uniformly distributed random linear hyperplanes in $\mathbb {R}^{d+1}$ , weakly converges to the typical cell of a stationary and isotropic Poisson hyperplane tessellation in $\mathbb {R}^d$ , as $n\to \infty $ .

The Gaussian polytope $\mathcal P_{n,d}$ is the convex hull of n independent standard normally distributed points in $\mathbb{R}^d$ . We derive explicit expressions for the probability that $\mathcal P_{n,d}$ contains a fixed point $x\in\mathbb{R}^d$ as a function of the Euclidean norm of x, and the probability that $\mathcal P_{n,d}$ contains the point $\sigma X$ , where $\sigma\geq 0$ is constant and X is a standard normal vector independent of $\mathcal P_{n,d}$ . As a by-product, we also compute the expected number of k-faces and the expected volume of $\mathcal P_{n,d}$ , thus recovering the results of Affentranger and Schneider (Discr. and Comput. Geometry, 1992) and Efron (Biometrika, 1965), respectively. All formulas are in terms of the volumes of regular spherical simplices, which, in turn, can be expressed through the standard normal distribution function $\Phi(z)$ and its complex version $\Phi(iz)$ . The main tool used in the proofs is the conic version of the Crofton formula.

Let $X,Y$ be two Hilbert spaces, let E be a subset of $X,$ and let $G\colon E \to Y$ be a Lipschitz mapping. A famous theorem of Kirszbraun’s states that there exists $\tilde {G} : X \to Y$ with $\tilde {G}=G$ on E and $ \operatorname {\mathrm {Lip}}(\tilde {G})= \operatorname {\mathrm {Lip}}(G).$ In this note we show that in fact the function $\tilde {G}:=\nabla _Y( \operatorname {\mathrm {conv}} (g))( \cdot , 0)$ , where

$$\begin{align*}g(x,y) = \inf_{z \in E} \Big\lbrace \langle G(z), y \rangle + \frac{\operatorname{\mathrm{Lip}}(G)}{2} \|(x-z,y)\|^2 \Big\rbrace + \frac{\operatorname{\mathrm{Lip}}(G)}{2}\|(x,y)\|^2, \end{align*}$$

$C^{1,1}$

All SL( $n$ ) contravariant symmetric matrix valued valuations on convex polytopes in $\mathbb{R}^{n}$ are completely classified without any continuity assumptions. The general Lutwak–Yang–Zhang matrix is shown to be essentially the unique such valuation.

Given a stationary and isotropic Poisson hyperplane process and a convex body K in ${\mathbb R}^d$ , we consider the random polytope defined by the intersection of all closed half-spaces containing K that are bounded by hyperplanes of the process not intersecting K. We investigate how well the expected mean width of this random polytope approximates the mean width of K if the intensity of the hyperplane process tends to infinity.

All non-negative, continuous, $\text{SL}(n)$, and translation invariant valuations on the space of super-coercive, convex functions on $\mathbb{R}^{n}$ are classified. Furthermore, using the invariance of the function space under the Legendre transform, a classification of non-negative, continuous, $\text{SL}(n)$, and dually translation invariant valuations is obtained. In both cases, different functional analogs of the Euler characteristic, volume, and polar volume are characterized.

This paper provides a functional analogue of the recently initiated dual Orlicz–Brunn–Minkowski theory for star bodies. We first propose the Orlicz addition of measures, and establish the dual functional Orlicz–Brunn–Minkowski inequality. Based on a family of linear Orlicz additions of two measures, we provide an interpretation for the famous $f$ -divergence. Jensen’s inequality for integrals is also proved to be equivalent to the newly established dual functional Orlicz–Brunn–Minkowski inequality. An optimization problem for the $f$ -divergence is proposed, and related functional affine isoperimetric inequalities are established.

The aim of this paper is to prove an inequality between relative entropy and the sum of average conditional relative entropies of the following form: for a fixed probability measure q on , ( is a finite set), and any probability measure on , (*)

$$D(p||q){\rm{\le}}C \cdot \sum\limits_{i = 1}^n {{\rm{\mathbb{E}}}_p D(p_i ( \cdot |Y_1 ,{\rm{ }}...,{\rm{ }}Y_{i - 1} ,{\rm{ }}Y_{i + 1} ,...,{\rm{ }}Y_n )||q_i ( \cdot |Y_1 ,{\rm{ }}...,{\rm{ }}Y_{i - 1} ,{\rm{ }}Y_{i + 1} ,{\rm{ }}...,{\rm{ }}Y_n )),} $$

The inequality (*) ismeaningful in product spaces, in both the discrete and the continuous case, and can be used to prove a logarithmic Sobolev inequality for q, provided uniform logarithmic Sobolev inequalities are available for qi(· |x1, …, xi−1, xi+1, …, xn), for all fixed i and fixed (x1, …, xi−1, xi+1, …, xn). Inequality (*) directly implies that the Gibbs sampler associated with q is a contraction for relative entropy.

In this paper we derive inequality (*), and thereby a logarithmic Sobolev inequality, in discrete product spaces, by proving inequalities for an appropriate Wasserstein-like distance.

In Weil (2001) formulae were proved for stationary Boolean models Z in ℝd with convex or polyconvex grains, which express the densities (specific mean values) of mixed volumes of Z in terms of related mean values of the underlying Poisson particle process X. These formulae were then used to show that in dimensions 2 and 3 the densities of mixed volumes of Z determine the intensity γ of X. For d = 4, a corresponding result was also stated, but the proof given was incomplete, since in the formula for the density of the Euler characteristic V̅0(Z) of Z a term $\overline V^{(0)}_{2,2}(X,X)$ was missing. This was pointed out in Goodey and Weil (2002), where it was also explained that a new decomposition result for mixed volumes and mixed translative functionals would be needed to complete the proof. Such a general decomposition result has recently been proved by Hug, Rataj, and Weil (2013), (2018) and is based on flag measures of the convex bodies involved. Here, we show that such flag representations not only lead to a correct derivation of the four-dimensional result, but even yield a corresponding uniqueness theorem in all dimensions. In the proof of the latter we make use of Alesker’s representation theorem for translation invariant valuations. We also discuss which shape information can be obtained in this way and comment on the situation in the nonstationary case.

For a fixed k ∈ {1, …, d}, consider arbitrary random vectors X0, …, Xk ∈ ℝd such that the (k + 1)-tuples (UX0, …, UXk) have the same distribution for any rotation U. Let A be any nonsingular d × d matrix. We show that the k-dimensional volume of the convex hull of affinely transformed Xi satisfies \[|{\rm{conv}}(A{X_{\rm{0}}} \ldots ,A{X_k}){\rm{|}}\mathop {\rm{ = }}\limits^{\rm{D}} (|{P_\xi }\varepsilon |/{\kappa _k})|{\rm{conv}}\left( {{X_0}, \ldots ,{X_k}} \right)\] , where ɛ:= {x ∈ ℝd : x┬ (A┬A)−1x ≤ 1} is an ellipsoid, Pξ denotes the orthogonal projection to a uniformly chosen random k-dimensional linear subspace ξ independent of X0, …, Xk, and κk is the volume of the unit k-dimensional ball. As an application, we derive the following integral geometry formula for ellipsoids: ck,d,p∫Ad,k |ɛ ∩ E|p+d+1μd,k(dE) = |ɛ|k+1∫Gd,k |PLɛ|pνd,k(dL), where $c_{k,d,p} = \big({\kappa_{d}^{k+1}}/{\kappa_k^{d+1}}\big) ({\kappa_{k(d+p)+k}}/{\kappa_{k(d+p)+d}})$ . Here p > −1 and Ad,k and Gd,k are the affine and the linear Grassmannians equipped with their respective Haar measures. The p = 0 case reduces to an affine version of the integral formula of Furstenberg and Tzkoni (1971).

Main results of the paper are as follows:

(1) For any finite metric space $M$ the Lipschitz-free space on $M$ contains a large well-complemented subspace that is close to $\ell _{1}^{n}$ .

(2) Lipschitz-free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell _{1}^{n}$ of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs.

Interesting features of our approach are: (a) We consider averages over groups of cycle-preserving bijections of edge sets of graphs that are not necessarily graph automorphisms. (b) In the case of such recursive families of graphs as Laakso graphs, we use the well-known approach of Grünbaum (1960) and Rudin (1962) for estimating projection constants in the case where invariant projections are not unique.

We describe the parameter spaces of some families of quadrilaterals, such as parallelograms, rectangles, rhombuses, cyclic quadrilaterals and trapezoids. For this purpose, we prove that the closed $n$ -disc $\mathbb{D}^{n}$ is the unique topological $n$ -manifold (with boundary) whose boundary and interior are homeomorphic to $\mathbb{S}^{n-1}$ and $\mathbb{R}^{n}$ , respectively. Roughly speaking, our main result states that the natural compactifications of the parameter spaces of cyclic quadrilaterals and of trapezoids, modulo similarity, are both homeomorphic to $\mathbb{D}^{3}$ .

We provide a unified approach that encompasses some integral formulas for functions of the visual angle of a compact convex set due to Crofton, Hurwitz and Masotti. The basic tool is an integral formula that also allows us to integrate new functions of the visual angle. Also, we establish some upper and lower bounds for the considered integrals, generalizing, in particular, those obtained by Santaló for Masotti’s integral.

Using previous results about shadow systems and Steiner symmetrization, we prove that the local maximizers of the volume product of convex bodies are actually the global maximizers, that is: ellipsoids.

A flag area measure on an $n$ -dimensional euclidean vector space is a continuous translation-invariant valuation with values in the space of signed measures on the flag manifold consisting of a unit vector $v$ and a $(p+1)$ -dimensional linear subspace containing $v$ with $0\leqslant p\leqslant n-1$ . Using local parallel sets, Hinderer constructed examples of $\text{SO}(n)$ -covariant flag area measures. There is an explicit formula for his flag area measures evaluated on polytopes, which involves the squared cosine of the angle between two subspaces. We construct a more general sequence of smooth $\text{SO}(n)$ -covariant flag area measures via integration over the normal cycle of appropriate differential forms. We provide an explicit description of our measures on polytopes, which involves an arbitrary elementary symmetric polynomial in the squared cosines of the principal angles between two subspaces. Moreover, we show that these flag area measures span the space of all smooth $\text{SO}(n)$ -covariant flag area measures, which gives a classification result in the spirit of Hadwiger’s theorem.

We consider the positive centre sets of regular $n$ -gons, rectangles and half discs, and conjecture a Bonnesen type inequality concerning positive centre sets which is stronger than the classical isoperimetric inequality.

The study of high-dimensional distributions is of interest in probability theory, statistics, and asymptotic convex geometry, where the object of interest is the uniform distribution on a convex set in high dimensions. The ℓp-spaces and norms are of particular interest in this setting. In this paper we establish a limit theorem for distributions on ℓp-spheres, conditioned on a rare event, in a high-dimensional geometric setting. As part of our proof, we establish a certain large deviation principle that is also relevant to the study of the tail behavior of random projections of ℓp-balls in a high-dimensional Euclidean space.