In this paper, we derive the exact formula for the probability that three randomly and uniformly selected points from the interior of the unit cube form vertices of an obtuse triangle.

We consider the asymptotic behaviour of the expectation of the perimeter deviation of a uniform random spherical disc–polygon in a spherical spindle convex disc with smooth boundary. We also introduce the notion of duality on the sphere, define a model of random circumscribed disc–polygons, and determine some asymptotic results about them.

This article studies the stability of the $L_p$ torsional measure in dimension $n \geq 3$. We show that the ball is the unique domain for the solution to the related overdetermined boundary value problem. Two concepts of relative asymmetry distance and $L^c$-distance have been used to estimate the stability of the $L_p$ torsional measure, based on different stability results of the $L_p$-width functionals. The result has been uniformly given via relative asymmetry distance for all $1<p \neq n+2$. In terms of $L^c$-distance, the result has been split into two cases: $p\in (1,n)$, and $n< p\neq n+2$.

We study a discrete process on planar convex bodies in which, at each step, a body is replaced by a weighted Minkowski average of itself and its rotation by a fixed angle. Up to translation and uniform scaling, this produces a rigid averaging dynamical system. We give a complete classification of the limit shapes. If the angle is an irrational multiple of $2\pi $, the iterates converge to a disk. If the angle is rational, they converge to the average of finitely many rotated copies of the initial body. We also obtain sharp convergence rates. In the rational case, the decay is uniform and exponential with an explicit constant depending only on the weight and the denominator of the angle. For irrational angles, we prove quantitative rates under a mild number-theoretic condition that holds for almost every angle: low regularity inputs have polynomial decay up to a logarithmic factor, while real analytic inputs have stretched exponential decay. For angles with bounded continued fraction coefficients, we give matching lower bounds along subsequences. These results describe the global attractors of the dynamics and indicate the absence of chaotic behaviour.

Inspired by the theory of hyperbolic polynomials and Hodge theory, we develop the theory of Lorentzian polynomials on cones. This notion captures the Hodge-Riemann relations of degree zero and one. Motivated by fundamental properties of volume polynomials of Chow rings of simplicial fans, we define a class of multivariate polynomials which we call hereditary polynomials. We give a complete and easily checkable characterization of hereditary Lorentzian polynomials. This characterization is used to give elementary and simple proofs of the Heron-Rota-Welsh conjecture for the characteristic polynomial of a matroid, and the Alexandrov-Fenchel inequalities for convex bodies.

We then characterize Chow rings of simplicial fans which satisfy the Hodge-Riemann relations of degree zero and one, and we prove that this property only depends on the support of the fan.

Several different characterizations of Lorentzian polynomials on cones are provided.

In this article, we reformulate LXYZ’s $L^p$ affine Sobolev inequality chain (including Lutwak–Yang–Zhang’s $L^p$ affine Sobolev inequality and Xiao’s p-affine capacity inequality) in the setting of Grassmann manifolds. For this purpose, the Grassmannian $ p $-affine capacity is introduced.

In this article, we study the $L_p$ Gauss image problem for C-pseudo-cones, where C is a pointed closed convex cone in $\mathbb {R}^n$ with vertex at the origin and having nonempty interior. We first establish the existence of solutions for a Borel measure supported on compact subsets, and then prove a general existence theorem for $p>0$ via approximation. In addition, we prove a uniqueness theorem for $p<0$ under appropriate assumptions.

In this article, we will provide a new proof of the fact that for any convex body $K\subseteq \mathbb {R}^n$,

$$ \begin{align*}\frac{{{2n}\choose{n}}}{n^n}n\int_0^\infty r^{n-1}\mathrm{vol}_n(K\cap(re_n+K))dr\leq\frac{(\mathrm{vol}_n(K))^{n+1}}{(\mathrm{vol}_{n-1}(P_{e_n^\perp}(K)))^n}, \end{align*} $$

$(e_i)_{i=1}^n$

$\mathbb {R}^n$

$P_{e_n^\perp }(K)$

$e_n$

$\mathrm {vol}_k$

Given $n$ convex bodies in the Euclidean space $\mathbb{R}^d$, we can find their volume polynomial which is a homogeneous polynomial of degree $d$ in $n$ variables. We consider the set of homogeneous polynomials of degree $d$ in $n$ variables that can be represented as the volume polynomial of any such given convex bodies. This set is a subset of the set of Lorentzian polynomials. Using our knowledge of operations that preserve the Lorentzian property, we give a complete classification of the cases for $(n,d)$ when the two sets are equal.

We define a spherical and hyperbolic analog to the Euclidean projection body for star bodies via the gnomonic projection from the unit sphere and stereographic projection in the hyperbolid model of hyperbolic space. We then prove a spherical and hyperbolic projection inequality for these notions by using an adaption of Steiner symmetrization for spherical, respectively, hyperbolic, star bodies.

We prove two-sided bounds on the expected values of several geometric functionals of the convex hull of Brownian motion in $\mathbb {R}^n$ and their inverse processes. This extends some recent results of McRedmond and Xu (2017), Jovalekić (2021), and Cygan, Šebek, and the first author (2023) from the plane to higher dimensions. Our main result shows that the average time required for the convex hull in $\mathbb {R}^n$ to attain unit volume is at most $n\sqrt [n]{n!}$. The proof relies on a novel procedure that embeds an n-simplex of prescribed volume within the convex hull of the Brownian path run up to a certain stopping time. All of our bounds capture the correct order of asymptotic growth or decay in the dimension n.

Averaged operators are important in Convex Analysis and Optimization Algorithms. In this article, we propose classifications of averaged operators, firmly nonexpansive operators, and proximal operators using the Bauschke–Bendit–Moursi modulus of averagedness. We show that if an operator is averaged with a constant less than $1/2$, then it is a bi-Lipschitz homeomorphism. Amazingly the proximal operator of a convex function has its modulus of averagedness less than $1/2$ if and only if the function is Lipschitz smooth. Some results on the averagedness of operator compositions are obtained. Explicit formulae for calculating the modulus of averagedness of resolvents and proximal operators in terms of various values associated with the maximally monotone operator or subdifferential are also given. Examples are provided to illustrate our results.

Nakamura and Tsuji recently obtained an integral inequality involving a Laplace transform of even functions that implies, at the limit, the Blaschke-Santaló inequality in its functional form. Inspired by their method, based on the Fokker-Planck semi-group, we extend the inequality to non-even functions. We consider a well-chosen centering procedure by studying the infimum over translations in a double Laplace transform. This requires a new look on the existing methods and leads to several observations of independent interest on the geometry of the Laplace transform. Application to reverse hypercontractivity is also given.

Let C be a convex body (i.e., a proper closed convex subset with nonempty interior) in a normed space X. We consider four moduli of local uniform rotundity for C at a given point $x\in \partial C$, that extend in a natural way the corresponding notions for the unit ball $B_X$, and we prove that they all coincide. This extends a known result of J. Daneš from 1976 concerning the particular case when $C=B_X$.

Let $H_n$ be the minimal number such that any n-dimensional convex body can be covered by $H_n$ translates of the interior of that body. Similarly $H_n^s$ is the corresponding quantity for symmetric bodies. It is possible to define $H_n$ and $H_n^s$ in terms of illumination of the boundary of the body using external light sources, and the famous Hadwiger’s covering conjecture (illumination conjecture) states that $H_n=H_{n}^s=2^n$. In this note, we obtain new upper bounds on $H_n$ and $H_{n}^s$ for small dimensions n. Our main idea is to cover the body by translates of John’s ellipsoid (the inscribed ellipsoid of the largest volume). Using specific lattice coverings, estimates of quermassintegrals for convex bodies in John’s position, and calculations of mean widths of regular simplexes, we prove the following new upper bounds on $H_n$ and $H_n^s$: $H_5\le 933$, $H_6\le 6137$, $H_7\le 41377$, $H_8\le 284096$, $H_4^s\le 72$, $H_5^s\le 305$, and $H_6^s\le 1292$. For larger n, we describe how the general asymptotic bounds $H_n\le \binom {2n}{n}n(\ln n+\ln \ln n+5)$ and $H_n^s\le 2^n n(\ln n+\ln \ln n+5)$ due to Rogers, Shephard and Roger, Zong, respectively, can be improved for specific values of n.

Gaussian random polytopes have received a lot of attention, especially in the case where the dimension is fixed and the number of points goes to infinity. Our focus is on the less-studied case where the dimension goes to infinity and the number of points is proportional to the dimension d. We study several natural quantities associated with Gaussian random polytopes in this setting. First, we show that the expected number of facets is equal to $C(\alpha)^{d+o(d)}$, where $C(\alpha)$ is some constant which depends on the constant of proportionality $\alpha$. We also extend this result to the expected number of k-facets. We then consider the more difficult problem of the asymptotics of the expected number of pairs of estranged facets of a Gaussian random polytope. When the number of points is 2d, we determine the constant C such that the expected number of pairs of estranged facets is equal to $C^{d+o(d)}$.

A convex body R in the hyperbolic plane is called reduced if any convex body $K\subset R$ has a smaller minimal width than R. We answer a few of Lassak’s questions about ordinary reduced polygons regarding its perimeter, diameter, and circumradius, and we also obtain a hyperbolic extension of a result of Fabińska.

A spectral convex set is a collection of symmetric matrices whose range of eigenvalues forms a symmetric convex set. Spectral convex sets generalize the Schur-Horn orbitopes studied by Sanyal–Sottile–Sturmfels (2011). We study this class of convex bodies, which is closed under intersections, polarity and Minkowski sums. We describe orbits of faces and give a formula for their Steiner polynomials. We then focus on spectral polyhedra. We prove that spectral polyhedra are spectrahedra and give small representations as spectrahedral shadows. We close with observations and questions regarding hyperbolicity cones, polar convex bodies and spectral zonotopes.

Let $\mathbb{P}_\kappa(n)$ be the probability that n points $z_1,\ldots,z_n$ picked uniformly and independently in $\mathfrak{C}_\kappa$, a regular $\kappa$-gon with area 1, are in convex position, that is, form the vertex set of a convex polygon. In this paper, we compute $\mathbb{P}_\kappa(n)$ up to asymptotic equivalence, as $n\to+\infty$, for all $\kappa\geq 3$, which improves on a famous result of Bárány (Ann. Prob. 27, 1999). The second purpose of this paper is to establish a limit theorem which describes the fluctuations around the limit shape of an n-tuple of points in convex position when $n\to+\infty$. Finally, we give an asymptotically exact algorithm for the random generation of $z_1,\ldots,z_n$, conditioned to be in convex position in $\mathfrak{C}_\kappa$.

We prove a functional version of the additive kinematic formula as an application of the Hadwiger theorem on convex functions together with a Kubota-type formula for mixed Monge–Ampère measures. As an application, we give a new explanation for the equivalence of the representations of functional intrinsic volumes as singular Hessian valuations and as integrals with respect to mixed Monge–Ampère measures. In addition, we obtain a new integral geometric formula for mixed area measures of convex bodies, where integration on $\operatorname {SO}(n-1)\times \operatorname {O}(1)$ is considered.