Related papers: An Isodiametric Problem with Additional Constraint…
This paper determines the optimal upper bound for the simultaneous packing and covering constants of the two-dimensional centrally symmetric convex domains. It solved a problem opening for more than thirty years.
We prove a sharp upper bound on convex domains, in terms of the diameter alone, of the best constant in a class of weighted Poincar\'e inequalities. The key point is the study of an optimal weighted Wirtinger inequality.
In this paper we prove: if a bounded domain with $C^2$ boundary covers a manifold which has finite volume with respect to either the Bergman volume, the K\"ahler-Einstein volume, or the Kobayashi-Eisenman volume, then the domain is…
In this paper, we characterize round spheres in the Euclidean space under some suitable conditions on the r-mean curvature.
We state that any constant curvature Riemannian metric with conical singularities of constant sign curvature on a compact (orientable) surface $S$ can be realized as a convex polyhedron in a Riemannian or Lorentzian) space-form. Moreover…
We discover some very general configuration results for constructing area-minimizing cones. In particular, given any closed minimal submanifold in some Euclidean sphere, every cone over the minimal product of sufficiently many copies of the…
We study the geometric significance of Leinster's notion of magnitude for a compact metric space. For a smooth, compact domain in an odd-dimensional Euclidean space, we show that the asymptotic expansion of the magnitude function at…
We provide a sufficient condition for the continuous extension of isometries for the Kobayashi distance between bounded convex domains in complex Euclidean spaces having boundaries that are only slightly more regular than $\mathcal{C}^1$.…
When a convex perfectly conducting inclusion is closely spaced to the boundary of the matrix domain, a bigger convex domain containing the inclusion, the electric field can be arbitrary large. We establish both the pointwise upper bound and…
We present a software suite for the analysis and optimization of ideal convex polyhedra in hyperbolic 3-space $\mathbb{H}^3$. Using Rivin's variational characterization of ideal polyhedra, we develop efficient algorithms for checking…
We raise and investigate the following problem that one can regard as a very close relative of the densest sphere packing problem. If the Euclidean 3-space is partitioned into convex cells each containing a unit ball, how should the shapes…
We show that the boundary of any bounded strongly pseudoconvex complete circular domain in $\mathbb C^2$ must contain points that are exceptionally tangent to a projective image of the unit sphere.
We consider a compact hyperbolic antiprism. It is a convex polyhedron with $2n$ vertices in the hyperbolic space $\mathbb{H}^3$. This polyhedron has a symmetry group $S_{2n}$ generated by a mirror-rotational symmetry of order $2n$, i.e.…
The second named author and David Kalaj introduced a pseudometric on any domain in the real Euclidean space $\mathbb R^n$, $n\ge 3$, defined in terms of conformal harmonic discs, by analogy with Kobayashi's pseudometric on complex…
The largest discs contained in a regular tetrahedron lie in its faces. The proof is closely related to the theorem of Fritz John characterising ellipsoids of maximal volume contained in convex bodies.
We show that, the solutions of the isoperimetric problem for small volumes are $C^{2,\alpha}$-close to small spheres. On the way, we define a class of submanifolds called pseudo balls, defined by an equation weaker than constancy of mean…
We prove some uniqueness results for conics of minimal area that enclose a compact, full-dimensional subset of the elliptic plane. The minimal enclosing conic is unique if its center or axes are prescribed. Moreover, we provide sufficient…
We show that the simplicial volume of a contractible 3-manifold not homeomorphic to $\mathbb{R}^3$ is infinite. As a consequence, the Euclidean space may be characterized as the unique contractible $3$-manifold with vanishing minimal…
The purpose of this paper is to provide a detailed description of the spaces that can be specified as $L^2$ domains for the operators of a first order elliptic complex on a compact manifold with conical singularities. This entails an…
The reverse isoperimetric problem asks for existence and properties of bounded convex sets in a Riemannian manifold which maximise the perimeter under all those sets of fixed volume which roll freely in a ball of some given radius. If the…