51 research outputs found
Local extrema for hypercube sections
Consider the hyperplanes at a fixed distance from the center of the
hypercube . Significant attention has been given to determining the
hyperplanes among these such that the -dimensional volume of
is maximal or minimal. In the spirit of a question by Vitali
Milman, the corresponding local problem is considered here when is
orthogonal to a diagonal or a sub-diagonal of the hypercube. It is proven in
particular that this volume is strictly locally maximal at the diagonals in all
dimensions greater than within a range for that grows like
. At lower order sub-diagonals, this volume is shown to be
strictly locally maximal when is close to and not locally extremal when
is large. This relies on a characterisation of local extremality at the
diagonals and sub-diagonals that allows to solve the problem over the whole
possible range for in any fixed, reasonably low dimension.Comment: 38 page
Once punctured disks, non-convex polygons, and pointihedra
We explore several families of flip-graphs, all related to polygons or
punctured polygons. In particular, we consider the topological flip-graphs of
once-punctured polygons which, in turn, contain all possible geometric
flip-graphs of polygons with a marked point as embedded sub-graphs. Our main
focus is on the geometric properties of these graphs and how they relate to one
another. In particular, we show that the embeddings between them are strongly
convex (or, said otherwise, totally geodesic). We also find bounds on the
diameters of these graphs, sometimes using the strongly convex embeddings.
Finally, we show how these graphs relate to different polytopes, namely type D
associahedra and a family of secondary polytopes which we call pointihedra.Comment: 24 pages, 6 figure
Flip-graph moduli spaces of filling surfaces
This paper is about the geometry of flip-graphs associated to triangulations
of surfaces. More precisely, we consider a topological surface with a
privileged boundary curve and study the spaces of its triangulations with n
vertices on the boundary curve. The surfaces we consider topologically fill
this boundary curve so we call them filling surfaces. The associated
flip-graphs are infinite whenever the mapping class group of the surface (the
group of self-homeomorphisms up to isotopy) is infinite, and we can obtain
moduli spaces of flip-graphs by considering the flip-graphs up to the action of
the mapping class group. This always results in finite graphs and we are
interested in their geometry.
Our main focus is on the diameter growth of these graphs as n increases. We
obtain general estimates that hold for all topological types of filling
surface. We find more precise estimates for certain families of filling
surfaces and obtain asymptotic growth results for several of them. In
particular, we find the exact diameter of modular flip-graphs when the filling
surface is a cylinder with a single vertex on the non-privileged boundary
curve.Comment: 52 pages, 29 figure
The Flip-Graph of the 4-Dimensional Cube is Connected
Flip-graph connectedness is established here for the vertex set of the 4-dimensional cube. It is found as a consequence that this vertex set has 92,487,256 triangulations, partitioned into 247,451 symmetry classe
The diameter of associahedra
It is proven here that the diameter of the d-dimensional associahedron is
2d-4 when d is greater than 9. Two maximally distant vertices of this polytope
are explicitly described as triangulations of a convex polygon, and their
distance is obtained using combinatorial arguments. This settles two problems
posed about twenty-five years ago by Daniel Sleator, Robert Tarjan, and William
Thurston.Comment: 28 pages, 14 figures, minor improvement
Weakly Regular Subdivisions
It is shown that 2-dimensional subdivisions can be made regular by moving their vertices within parallel 1-dimensional spaces. As a consequence, any 2-dimensional subdivision is projected from the boundary complex of a 4-polytop
The flip-graph of the 4-dimensional cube is connected
Flip-graph connectedness is established here for the vertex set of the
4-dimensional cube. It is found as a consequence that this vertex set has 92
487 256 triangulations, partitioned into 247 451 symmetry classes.Comment: 20 pages, 3 figures, revised proofs and notation
Many equiprojective polytopes
A -dimensional polytope is -equiprojective when the projection of
along any line that is not parallel to a facet of is a polygon with
vertices. In 1968, Geoffrey Shephard asked for a description of all
equiprojective polytopes. It has been shown recently that the number of
combinatorial types of -equiprojective polytopes is at least linear as a
function of . Here, it is shown that there are at least such
combinatorial types as goes to infinity. This relies on the
Goodman--Pollack lower bound on the number of order types and on new
constructions of equiprojective polytopes via Minkowski sums.Comment: 22 pages, 3 figure
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