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Joël Rouyer

Publications and source records attributed to Joël Rouyer.

10 recordsLinked to original sources
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

Steinhaus conditions for convex polyhedra

On a convex surface $S$, the antipodal map $F$ associates to a point $p$ the set of farthest points from $p$, with respect to the intrinsic metric. $S$ is called a Steinhaus surface if $F$ is a single-valued involution. We prove that any convex polyhedron has an open and dense set of points $p$ admitting a unique antipode $F_p$, which in turn admits a unique antipode $F_{F_p}$, distinct from $p$. In particular, no convex polyhedron is Steinhaus.

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↗
arXiv2010

Generic Properties of Compact Metric Spaces

We prove that there is a residual subset of the Gromov-Hausdorff space (i.e. the space of all compact metric spaces up to isometry endowed with the Gromov-Hausdorff distance) whose points enjoy several unexpected properties. In particular, they have zero lower box dimension and infinite upper box dimension.

math.MG↗