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Costin Vîlcu

Publications and source records attributed to Costin Vîlcu.

18 recordsLinked to original sources
arXiv2021

Cut Locus Realizations on Convex Polyhedra

We prove that every positively-weighted tree T can be realized as the cut locus C(x) of a point x on a convex polyhedron P, with T weights matching C(x) lengths. If T has n leaves, P has (in general) n+1 vertices. We show there are in fact a continuum of polyhedra P each realizing T for some x on P. Three main tools in the proof are properties of the star unfolding of P, Alexandrov's gluing theorem, and a cut-locus partition lemma. The construction of P from T is surprisingly simple.

cs.CG↗
arXiv2019

Locating diametral points

Let $K$ be a convex body in $\mathbb{R} ^d$, with $d = 2,3$. We determine sharp sufficient conditions for a set $E$ composed of $1$, $2$, or $3$ points of ${\rm bd}K$, to contain at least one endpoint of a diameter of $K$ (for $d=2,3$). We extend this also to convex surfaces, with their intrinsic metric. Our conditions are upper bounds on the sum of the complete angles at the points in $E$. We also show that such criteria do not exist for $n\geq 4$ points.

math.MG↗
arXiv2019

With respect to whom are you critical?

For any compact Riemannian surface $S$ and any point $y$ in $S$, $Q_y^{-1}$ denotes the set of all points in $S$, for which $y$ is a critical point. We proved \cite{BIVZ} together with Imre Bárány that card$Q_y^{-1} \geq 1$, and that equality for all $y\in S$ characterizes the surfaces homeomorphic to the sphere. Here we show, for any orientable surface $S$ and any point $y \in S$, the following two main results. There exist an open and dense set of Riemannian metrics $g$ on $S$ for which $y$ is critical with respect to an odd number of points in $S$, and this is sharp. Card$Q_y^{-1} \leq 5$ for the torus and card$Q_y^{-1} \leq 8g-5$ if the genus $g$ of $S$ is at least $2$. Properties involving points at globally maximal distance on $S$ are eventually presented.

math.GT↗
arXiv2018

Farthest points on flat surfaces

We consider the distance function from an arbitrary point $p$ on a flat surface, and determine the set $F_{p}$ of all \emph{farthest points} (i.e., points at maximal distance) from $p$.

math.MG↗
arXiv2018

Double normals of most convex bodies

We consider a typical (in the sense of Baire categories) convex body $K$ in $\mathbb{R}^{d+1}$. The set of feet of its double normals is a Cantor set, having lower box-counting dimension $0$ and packing dimension $d$. The set of lengths of those double normals is also a Cantor set of lower box-counting dimension $0$. Its packing dimension is equal to $\frac{1}{2}$ if $d=1$, is at least $\frac{3}{4}$ if $d=2$, and equals $1$ if $d\geq3$. We also consider the lower and upper curvatures at feet of double normals of $K$, with a special interest for local maxima of the length function (they are countable and dense in the set of double normals). In particular, we improve a previous result about the metric diameter.

math.MG↗
arXiv2015

Envelopes of $α$-sections

Let $K$ be a planar convex body af area $|K|$, and take $0 \textless{} α\textless{} 1$.An $α$-section of $K$ is a line cutting $K$ into two parts, one of whichhas area $α|K|$. This article presents a systematic study of the envelope of $α$-sectionsand its dependence on $α$. Several open questions are asked, one of themin relation to a problem of fair partitioning.

math.MG↗
arXiv2013

Simple closed geodesics on most Alexandrov surfaces

We study the existence of simple closed geodesics on most (in the sense of Baire category) Alexandrov surfaces with curvature bounded below, compact and without boundary. We show that it depends on both the curvature bound and the topology of the surfaces.

math.MG↗
arXiv2013

The connected components of the space of Alexandrov surfaces

Denote by $\mathcal{A}(κ)$ the set of all compact Alexandrov surfaces with curvature bounded below by $κ$ without boundary, endowed with the topology induced by the Gromov-Hausdorff metric. We determine the connected components of $\mathcal{A}(κ)$ and of its closure.

math.MG↗
arXiv2013

Every graph is a cut locus

We prove that every connected graph can be realized as the cut locus of some point on some Riemannian surface $S$ which, in some cases, has constant curvature. We study the stability of such realizations, and their generic behavior.

math.DG↗
arXiv2011

Orientable cut locus structures on graphs

We showed in another paper [arXiv:1103.1759] that every connected graph can be realized as the cut locus of some point on some riemannian surface $S$. Here, criteria for the orientability of $S$ are given, and are applied to classify the distinct, orientable, cut locus structures on graphs with four generating cycles.

math.DG↗
arXiv2011

Cut locus structures on graphs

Motivated by a fundamental geometrical object, the cut locus, we introduce and study a new combinatorial structure on graphs.

cs.DM↗
arXiv2011

On the number of cut locus structures on graphs

We proved in another paper that every connected graph can be realized as the cut locus of some point on some riemannian surface. Here we give upper bounds on the number of such realizations.

math.CO↗
arXiv2009

Star Unfolding Convex Polyhedra via Quasigeodesic Loops

We extend the notion of star unfolding to be based on a quasigeodesic loop Q rather than on a point. This gives a new general method to unfold the surface of any convex polyhedron P to a simple (non-overlapping), planar polygon: cut along one shortest path from each vertex of P to Q, and cut all but one segment of Q.

cs.CG↗
arXiv2008

Unfolding Convex Polyhedra via Quasigeodesic Star Unfoldings

We extend the notion of a star unfolding to be based on a simple quasigeodesic loop Q rather than on a point. This gives a new general method to unfold the surface of any convex polyhedron P to a simple, planar polygon: shortest paths from all vertices of P to Q are cut, and all but one segment of Q is cut.

cs.CG↗