Related papers: The p-widths of the Hemisphere

We define and study generalizations of simplicial volume over arbitrary seminormed rings with a focus on $p$-adic simplicial volumes. We investigate the dependence on the prime and establish homology bounds in terms of $p$-adic simplicial…

We show that round hemispheres are the only compact 2 dimensional Riemannian manifolds (with or without boundary) such that almost every pair of complete geodesics intersect once and only once. We prove this by establishing a sharp…

In this paper we study $L_p$-norm spherical copulas for arbitrary $p \in [1,\infty]$ and arbitrary dimensions. The study is motivated by a conjecture that these distributions lead to a sharp bound for the value of a certain generalized mean…

We prove an $\epsilon$-regularity theorem for vector-valued p-harmonic maps, which are critical with respect to a partially free boundary condition, namely that they map the boundary into a round sphere. This does not seem to follow from…

We construct a surface with a cylindrical end which has a finite number of Laplace eigenvalues embedded in its continuous spectrum. The surface is obtained by attaching a cylindrical end to a hyperbolic torus with a hole. To our knowledge,…

We characterize the standard $\mathbb{S}^3$ as the closed Ricci-positive 3-manifold with scalar curvature at least 6 having isoperimetric surfaces of largest area: $4\pi$. As a corollary we answer in the affirmative an interesting special…

We determine all non-edge-to-edge tilings of the sphere by regular spherical polygons of three or more sides.

In this paper, free boundary geodesic networks whose length realize the first min-max width of the length functional are investigated. This functional acts on the space of relative flat 1-dimensional cycles modulo 2 in a compact surface…

In this survey article, we discuss some recent progress on geometric analysis on manifold with ends. In the final section, we construct manifolds with ends with oscillating volume functions which may turn out to have a different heat kernel…

A zone of width $\omega$ on the unit sphere is the set of points within spherical distance $\omega/2$ of a given great circle. We show that the total width of any collection of zones covering the unit sphere is at least $\pi$, answering a…

We explore a definition of uniformity on noncompact manifolds that does not require a Riemannian metric, but is equivalent to bounded gemetry. These are unfinished research notes (and will likely never be published), but since they were…

Antonio Ros gave a lower bound for the first eigenvalue $\lambda_1$ of $\Delta$ of a $P$-manifold $(M, g)$ in terms of the lower bound on the Ricci curvature $Ric_M$ and asked what happened when this lower bound was achieved. In this paper…

The theory of boundary regularity for $p$-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a $p$-Poincar\'e inequality, $1<p<\infty$. The barrier classification of regular…

The study of the graph diameter of polytopes is a classical open problem in polyhedral geometry and the theory of linear optimization. In this paper we continue the investigation initiated in [4] by introducing a vast hierarchy of…

We extend the $L^4$-square function estimates for the parabola and the half-cone to quadratic manifolds in higher dimensions and their conical extensions. To this end, we require transversality for the tangent spaces of the quadratic…

We consider the problem of determining the manifold $n$-widths of Sobolev and Besov spaces with error measured in the $L_p$-norm. The manifold widths control how efficiently these spaces can be approximated by general non-linear parametric…

We determine an upper bound for the size of a stem $p$-group in terms of its breadth and its maximum character degree.

The notion of the Urysohn $d$-width measures to what extent a metric space can be approximated by a $d$-dimensional simplicial complex. We investigate how local Urysohn width bounds on a riemannian manifold affect its global width. We bound…

For a given convex body K in $R^d$, a random polytope $K^{(n)}$ is defined (essentially) as the intersection of $n$ independent closed halfspaces containing $K$ and having an isotropic and (in a specified sense) uniform distribution. We…

Motivated by Leinster-Cobbold measures of biodiversity, the notion of the spread of a finite metric space is introduced. This is related to Leinster's magnitude of a metric space. Spread is generalized to infinite metric spaces equipped…