arXiv subjects

Ásgeir Valfells

Publications and source records attributed to Ásgeir Valfells.

2 recordsLinked to original sources

arXiv2023

Minimizing edge-length polyhedrons

A 1957 conjecture by Zdzislaw Melzak, that the unit volume polyhedron with least edge length was a triangular right prism, with edge length $2^{2/3}3^{11/6}$. We present a variety of necessary local criteria for any minimizer. In the case that we are restricted to convex polyhedrons we demonstrate that all vertices must be of degree three, the number of triangular faces is at most 14, and we describe the behavior of quadrilateral faces should they become arbitrarily small.

math.MG↗

arXiv2023

Minimal Piecewise Linear Cones in $\mathbb{R}^{4}$

We consider three dimensional piecewise linear cones in $\mathbb{R}^4$ that are mass minimizing w.r.t. Lipschitz maps in the sense of \cite{almgren1976existence} as in \cite{Taylor76}. There are three that arise naturally by taking products of $\mathbb{R}$ with lower dimensional cases and earlier literature has demonstrated the existence of two with 0-dimensional singularities. We classify all possible candidates and demonstrate that there are no p.l. minimizers outside these five.

math.DG↗