Related papers: New bounds for the Heilbronn triangle problem

We study the time complexity of the discrete $k$-center problem and related (exact) geometric set cover problems when $k$ or the size of the cover is small. We obtain a plethora of new results: - We give the first subquadratic algorithm for…

We study the minimal surface equation in the Heisenberg space, Nil_3. A geometric proof of non existence of minimal graphs over non convex, bounded and unbounded domains is achieved (our proof holds in the Euclidean space as well). We solve…

The present manuscript aims to derive an expression for the lower bound of the modulus of the Dirichlet eta function on vertical lines $\Re(s)=\alpha$. The approach employs concepts of two-dimensional principal component analysis built on a…

In this paper, we consider the obstacle problem for the fractional Laplace operator $(-\Delta)^s$ in the Euclidian space $\mathbb{R}^n$ in the case where $1<s<2$. As first observed in \cite{Y}, the problem can be extended to the upper…

We show that every planar convex body is contained in a quadrangle whose area is less than $(1 - 2.6 \cdot 10^{-7}) \sqrt{2}$ times the area of the original convex body, improving the best known upper bound by W. Kuperberg.

We present a new proof (based on spectral decomposition) of a bound originally proved by Sidelnikov~\, for the frame potentials $\sum_{ij} \left( {\bf P}_i \cdot {\bf P}_j \right)^\ell $ on a unit--sphere in $d$ dimensions. Sidelnikov's…

A triangulation of a surface is \emph{irreducible} if there is no edge whose contraction produces another triangulation of the surface. We prove that every irreducible triangulation of a surface with Euler genus $g\geq1$ has at most $13g-4$…

Given a set $\mathcal{P}$ of $h$ pairwise disjoint simple polygonal obstacles in $\mathbb{R}^2$ defined with $n$ vertices, we compute a sketch $\Omega$ of $\mathcal{P}$ whose size is independent of $n$, depending only on $h$ and the input…

We study the problem of computing the minimum area triangle that circumscribes a given $n$-sided convex polygon touching edge-to-edge. In other words, we compute the minimum area triangle that is the intersection of 3 half-planes out of $n$…

We prove rigidity results involving the Hawking mass for surfaces immersed in a $3$-dimensional, complete Riemannian manifold $(M,g)$ with non-negative scalar curvature (resp. with scalar curvature bounded below by $-6$). Roughly, the main…

The problem widely known as Moser's Square Packing Problem asks for the smallest area $A$ such that for any set $S$ of squares of total area $1$, there exists a rectangle $R$ of area $A$ into which the squares in $S$ permit an…

Let $B$ be a compact Riemann surface and $B_0\subset B$ a bordered hyperbolic subsurface obtained by removing finitely many disjoint closed disks. Fix a nontrivial loop $\alpha$ in $B_0$. For $s\ge 0$, let $L(\alpha,s)$ denote the supremum,…

The metric sketching problem is defined as follows. Given a metric on $n$ points, and $\epsilon>0$, we wish to produce a small size data structure (sketch) that, given any pair of point indices, recovers the distance between the points up…

On a finite-volume hyperbolic $3$-manifold, we establish an upper bound on the area of closed embedded surfaces with constant mean curvature at least one, depending on the mean curvature and the genus bounds. This area bound implies…

We present a technique for deriving lower bounds for incidences with hypersurfaces in ${\mathbb R}^d$ with $d\ge 4$. These bounds apply to a large variety of hypersurfaces, such as hyperplanes, hyperspheres, paraboloids, and hypersurfaces…

This paper gives a framework to produce the lower bound of eigenvalues defined in a Hilbert space by the eigenvalues defined in another Hilbert space. The method is based on using the max-min principle for the eigenvalue problems.

We obtain bounds for the Faltings's delta function for any Riemann surface of genus greater than one. The bounds are in terms of the genus of the surface and two basic quantities coming from hyperbolic geometry: The length of the shortest…

We study the finite element approximation of the Kirchhoff plate equation on domains with curved boundaries using the Hellan-Herrmann-Johnson (HHJ) method. We prove optimal convergence on domains with piecewise $C^{k+1}$ boundary for $k…

Suppose that $M$ is a compact Riemannian manifold with boundary and $u$ is an $L^2$-normalized Dirichlet eigenfunction with eigenvalue $\lambda$. Let $\psi$ be its normal derivative at the boundary. Scaling considerations lead one to expect…

In this paper, we study closed embedded minimal hypersurfaces in a Riemannian $(n+1)$-manifold ($2\le n\le 6$) that minimize area among such hypersurfaces. We show they exist and arise either by minimization techniques or by min-max…