相关论文: Area-minimizing unit vector fields on some spheric…
While there is extensive literature on approximation of convex bodies by inscribed or circumscribed polytopes, much less is known in the case of generally positioned polytopes. Here we give upper and lower bounds for approximation of convex…
We determine the least-area unit-volume tetrahedral tile of Euclidean space, without the constraint of Gallagher et al. that the tiling uses only orientation-preserving images of the tile. The winner remains Sommerville's type 4v.
We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.
We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.
Inspired by the boolean discrepancy problem, we study the following optimization problem which we term \textsc{Spherical Discrepancy}: given $m$ unit vectors $v_1, \dots, v_m$, find another unit vector $x$ that minimizes $\max_i \langle x,…
We prove essentially optimal bounds for norms of spectral projectors on thin spherical shells for the Laplacian on the cylinder (R/Z)*R. In contrast to previous investigations into spectral projectors on tori, having one unbounded dimension…
We consider in this paper an area functional defined on submanifolds of fixed degree immersed into a graded manifold equipped with a Riemannian metric. Since the expression of this area depends on the degree, not all variations are…
It is proved by Brendle in [4] that the equatorial disk $D^k$ has least area among $k$-dimensional free boundary minimal surfaces in the Euclidean ball $B^n$. By comparing the excess of free boundary minimal surfaces with the excess of the…
We consider the minimal energy problem on the unit sphere $\mathbb S^2$ in the Euclidean space $\mathbb R^3$ immersed in an external field $Q$, where the charges are assumed to interact via Newtonian potential $1/r$, $r$ being the Euclidean…
We prove that the area of a free boundary minimal surface $\Sigma^2 \subset B^n$, where $B^n$ is a geodesic ball contained in a round hemisphere $\mathbb{S}^n_+$, is at least as big as that of a geodesic disk with the same radius as $B^n$;…
The aim of this paper is to give upper bounds for the Euclidean minima of abelian fields of odd prime conductor. In particular, these bounds imply Minkowski's conjecture for totally real number fields of conductor p^r, where p is an odd…
We write down a one-dimensional integral formula and compute large-n asymptotics for the expectation of the absolute value of the smallest component of a unit vector in n-dimensional Euclidean space. The method is general, and allows to…
We use the theory of calibrations to write the equation of a minimal volume vector field on a given Riemann surface.
We study the projections in vector spaces over finite fields. We prove finite fields analogues of the bounds on the dimensions of the exceptional sets for Euclidean projection mapping. We provide examples which do not have exceptional…
We establish upper bounds for the size of two-distance sets in Euclidean space and spherical two-distance sets. The main recipe for obtaining upper bounds is the spectral method. We construct Seidel matrices to encode the distance relations…
In this paper we prove that a flat free-boundary minimal $n$-disk, $n\geq3$, in the unit Euclidean ball $B^{n+1}$ is the unique compact free boundary minimal hypersurface in the unit Euclidean ball which the squared norm of the second…
We solve the problem of finding a sharp upper bound on the minimum angle formed by $N$ points in the Euclidean and Hyperbolic planes.
For a convex body on the Euclidean unit sphere the spherical convex floating body is introduced. The asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated. This gives rise to…
The issue of the existence and possible triviality of the Euclidean quantum scalar field in dimension 4 is investigated by using some large deviations techniques. As usual, the field $\varphi_{d}^{4}$ is obtained as a limit of regularized…
We give a sharp lower bound on the area of the domain enclosed by an embedded curve lying on a two-dimensional sphere, provided that geodesic curvature of this curve is bounded from below. Furthermore, we prove some dual inequalities for…