Related papers: Covering a reduced spherical body by a disk

In [7], Guan, Ren and Wang obtained a $C^2$ a priori estimate for admissible 2-convex hypersurfaces satisfying the Weingarten curvature equation $\sigma_2(\kappa(X))=f(X, \nu(X)).$ In this note, we give a simpler proof of this result, and…

A new method is developed which permits the reconstruction of the surface-density distribution in the galactic disk of finite radius from an arbitrary smooth distribution of the angular velocity via two simple quadratures. The existence of…

We consider convex sets whose modulus of convexity is uniformly quadratic. First, we observe several interesting relations between different positions of such ``2-convex'' bodies; in particular, the isotropic position is a finite…

We prove that for any convex polygon $S$ with at least four sides, or a concave one with no parallel sides, and any $m>0$, there is an $m$-fold covering of the plane with homothetic copies of $S$ that cannot be decomposed into two…

It is well known that the lattice packing density and the lattice covering density of a triangle are $\frac{2}{3}$ and $\frac{3}{2}$ respectively. We also know that the lattices that attain these densities both are unique. Let…

We confirm two conjectures of Lassak on the area of reduced spherical polygons. The area of every reduced spherical non-regular $n$-gon is less than that of the regular spherical $n$-gon of the same thickness. Moreover, the area of every…

By making use of arithmetic information inequalities, we give a strong quantitative bound for the discretised ring theorem. In particular, we show that if $A \subset [1,2]$ is a $(\delta,\sigma)$-set, with $|A| = \delta^{-\sigma},$ then…

Polytopes are the basic finite data structures for convex sets: they appear as feasible regions in linear optimization, as geometric summaries in algorithms, and as random objects in stochastic geometry. A natural geometric question is…

We prove a Ros-Rosenberg theorem in the setting of Special Weingarten surfaces. We show that a compact, connected, embedded, Special Weingarten surface in $\mathhb{R}^3$ with planar convex boundary is a topological disk under mild suitable…

We show that the rate of convergence on the approximation of volumes of a convex symmetric polytope P in R^n by its dual L_{p$-centroid bodies is independent of the geometry of P. In particular we show that if P has volume 1,…

We prove a qualitative and a quantitative stability of the following rigidity theorem: an anisotropic totally umbilical closed hypersurface is the Wulff shape. Consider $n \geq 2$, $p\in (1, \, +\infty)$ and $\Sigma$ an $n$-dimensional,…

We generalize the second pinching theorem for minimal hypersurfaces in a sphere due to Peng-Terng, Wei-Xu, Zhang, and Ding-Xin to the case of hypersurfaces with small constant mean curvature. Let $M^n$ be a compact hypersurface with…

It is shown that two strictly pseudoconvex Stein domains with real analytic boundaries have biholomorphic universal coverings provided that their boundaries are locally biholomorphically equivalent. This statement can be regarded as a…

Let $H_n$ be the minimal number of smaller homothetic copies of an $n$-dimensional convex body required to cover the whole body. Equivalently, $H_n$ can be defined via illumination of the boundary of a convex body by external light sources.…

A Minkowski class is a closed subset of the space of convex bodies in Euclidean space Rn which is closed under Minkowski addition and non-negative dilatations. A convex body in Rn is universal if the expansion of its support function in…

Minkowski's 2nd theorem in the Geometry of Numbers provides optimal upper and lower bounds for the volume of a $o$-symmetric convex body in terms of its successive minima. In this paper we study extensions of this theorem from two different…

A celebrated result of Beck shows that for any set of $N$ points on $\mathbb{S}^d$ there always exists a spherical cap $B \subset \mathbb{S}^d$ such that number of points in the cap deviates from the expected value $\sigma(B) \cdot N$ by at…

We extend Struwe's result (Acta Math., 1988) on the existence of free boundary constant mean curvature disks to almost every prescribed boundary contact angle in $(0, \pi)$. Specifically, let $\Sigma$ be a surface in $\mathbb{R}^3$…

We show that for every convex polyhedral sphere $P$ in $S^3$, there exist two canonical, non-edge-to-edge tilings of $S^{2}$ whose tiles are given by all the faces of $P$ and the dual convex polyhedral sphere $P^*$ to $P$. Under the…

Using Spitzer Space Telescope photometric observations of the eclipsing, interacting binary WZ Sge, we have discovered that the accretion disk is far more complex than previously believed. Our 4.5 and 8 micron time series observations…