Related papers: The Busemann-Petty Problem in Complex Hyperbolic S…
We determine the minimal volume of arithmetic hyperbolic orientable n-dimensional orbifolds (compact and non-compact) for every odd dimension n>3. Combined with the previously known results it solves the minimal volume problem for…
The 2-body problem on the sphere and hyperbolic space are both real forms of holomorphic Hamiltonian systems defined on the complex sphere. This admits a natural description in terms of biquaternions and allows us to address questions…
We prove several estimates for the moments of arbitrary measures on convex bodies. We apply these estimates to show a new slicing inequality for measures on convex bodies. We also deduce estimates for the outer volume ratio distance from an…
We study the existence and non-existence of positive solutions for the following class of nonlinear elliptic problems in the hyperbolic space $$ -\Delta_{\mathbb{B}^N} u-\lambda u=a(x)u^{p-1} \, + \, \varepsilon u^{2^*-1}…
A translation body of a convex body is the convex hull of two of its translates intersecting each other. In the 1950s, Rogers and Shephard found the extremal values, over the family of $n$-dimensional convex bodies, of the maximal volume of…
We establish a symmetry result for positive entire solutions with a prescribed growth rate to the following fourth order equation on the 3-dimensional hyperbolic space $\mathbb{H}^3$: \[ P_2 u = - u^{-7}, \] where $P_2$ denotes the…
The new result of this paper connected with the following problem: Consider a supporting hyperplane of a regular simplex and its re ected image at this hyperplane. When will be the volume of the convex hull of these two simplices maximal?…
Volume is a natural measure of complexity of a Riemannian manifold. In this survey, we discuss the results and conjectures concerning n-dimensional hyperbolic manifolds and orbifolds of small volume.
We model the universe as a 3-brane embedded in five dimensional spacetime with N=2 supersymmetry. The presence of the scalar fields of the universal hypermultiplet in the bulk results in a positive pressure effectively reducing the value of…
We prove that if all intersections of a convex body $B\subset\mathbb R^4$ with 3-dimensional linear subspaces are linearly equivalent then $B$ is a centered ellipsoid. This gives an affirmative answer to the case $n=3$ of the following…
We show that the renormalized volume of a quasifuchsian hyperbolic 3-manifold is equal, up to an additive constant, to the volume of its convex core. We also provide a precise upper bound on the renormalized volume in terms of the…
The main goal of this paper is to present a series of inequalities connecting the surface area measure of a convex body and surface area measure of its projections and sections. We present a solution of a question from S. Campi, P.…
A convex body $R$ in the hyperbolic plane is 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…
Let $\Omega$ be a measurable Euclidean set in $\mathbb{R}^{n}$ that is symmetric, i.e. $\Omega=-\Omega$, such that $\Omega\times\mathbb{R}$ has the smallest Gaussian surface area among all measurable symmetric sets of fixed Gaussian volume.…
The goal of this paper is to present a lower bound for the Mahler volume of at least 4-dimensional symmetric convex bodies. We define a computable dimension dependent constant through a 2-dimensional variational (max-min) procedure and…
In this note we investigate the behavior of the volume that the convex hull of two congruent and intersecting simplices in Euclidean $n$-space can have. We prove some useful equalities and inequalities on this volume. For the regular…
A new intrinsic volume metric is introduced for the class of convex bodies in $\mathbb{R}^n$. As an application, an inequality is proved for the asymptotic best approximation of the Euclidean unit ball by arbitrarily positioned polytopes…
We focus on the problems of existence and non-existence of positive solutions for the Sobolev-subcritical Lane-Emden equation on certain Riemannian manifolds (mainly models) with asymptotically negative curvature, which, from the viewpoint…
The quantum $N$-body problem is studied in the context of nonrelativistic quantum mechanics with a one-dimensional deformed Heisenberg algebra of the form $[\hat x,\hat p]=i(1+\beta \hat p^2)$, leading to the existence of a minimal…
We study the existence of sign-changing solutions with multiple bubbles to the slightly subcritical problem $$-\Delta u=|u|^{2^*-2-\e}u \hbox{in}\Omega, \quad u=0 \hbox{on}\partial \Omega,$$ where $\Omega$ is a smooth bounded domain in…