Stability of graphs.
Stratification of the discriminant in Reflection groups.
Sur deux suites récurrentes du second ordre, linéaires et homogènes
Sur les cercles tangents à trois cercles et les sphères tangentes à trois ou à quatre sphères
Symmetries of regular four-dimensional polytopes
Symmetries of woven fabrics
Systoles of a family of triangle surfaces.
Tangent Lines and Lipschitz Differentiability Spaces
We study the existence of tangent lines, i.e. subsets of the tangent space isometric to the real line, in tangent spaces of metric spaces.We first revisit the almost everywhere metric differentiability of Lipschitz continuous curves. We then show that any blow-up done at a point of metric differentiability and of density one for the domain of the curve gives a tangent line. Metric differentiability enjoys a Borel measurability property and this will permit us to use it in the framework of Lipschitz...
The Boundary at Infinity of a Rough CAT(0) Space
We develop the boundary theory of rough CAT(0) spaces, a class of length spaces that contains both Gromov hyperbolic length spaces and CAT(0) spaces. The resulting theory generalizes the common features of the Gromov boundary of a Gromov hyperbolic length space and the ideal boundary of a complete CAT(0) space. It is not assumed that the spaces are geodesic or proper
The elementary geometry of a triangular world with hexagonal circles.
The inner-product of iso-taxicab geometry.
The lattices and Möbius functions of stable closed subrootsystems and Hyperplane complements for classical Weyl Groups.
The length of a unitary transformation.
The length of a unitary transformation for characteristic 2.
The order of a finite Coxeter group.
The Rotation Group
We introduce length-preserving linear transformations of Euclidean topological spaces. We also introduce rotation which preserves orientation (proper rotation) and reverses orientation (improper rotation). We show that every rotation that preserves orientation can be represented as a composition of base proper rotations. And finally, we show that every rotation that reverses orientation can be represented as a composition of proper rotations and one improper rotation.
The Two Most Algebraic K3 Surfaces.
Theory of equidistance and betweenness relations in regular metric spaces
Topologische Möbiusebenen in spiegelungsgeometrischer Darstellung
Topology of the isometry group of the Urysohn space
Using classical results of infinite-dimensional geometry, we show that the isometry group of the Urysohn space, endowed with its usual Polish group topology, is homeomorphic to the separable Hilbert space ℓ²(ℕ). The proof is based on a lemma about extensions of metric spaces by finite metric spaces, which we also use to investigate (answering a question of I. Goldbring) the relationship, when A,B are finite subsets of the Urysohn space, between the group of isometries fixing A pointwise, the group...