49 research outputs found
Strictly convex drawings of planar graphs
Every three-connected planar graph with n vertices has a drawing on an O(n^2)
x O(n^2) grid in which all faces are strictly convex polygons. These drawings
are obtained by perturbing (not strictly) convex drawings on O(n) x O(n) grids.
More generally, a strictly convex drawing exists on a grid of size O(W) x
O(n^4/W), for any choice of a parameter W in the range n<W<n^2. Tighter bounds
are obtained when the faces have fewer sides.
In the proof, we derive an explicit lower bound on the number of primitive
vectors in a triangle.Comment: 20 pages, 13 figures. to be published in Documenta Mathematica. The
revision includes numerous small additions, corrections, and improvements, in
particular: - a discussion of the constants in the O-notation, after the
statement of thm.1. - a different set-up and clarification of the case
distinction for Lemma
Tverberg's theorem, a new proof
We give a new proof Tverberg's famous theorem: For every set
with , there is a partition of into sets
such that \bigcap_{p=1}^r \conv X_p\ne \emptyset. The new
proof uses linear algebra, specially structured matrices, the theory of linear
equations, and Tverberg's original ``moving the points" method
Positive bases, cones, Helly type theorems
Assume that is a positive integer and \C is a finite collection
of convex bodies in . We prove a Helly type theorem: If for every
subfamily \C^*\subset \C of size at most the set
\bigcap \C^* contains a -dimensional cone, then so does \bigcap \C. One
ingredient in the proof is another Helly type theorem about the dimension of
lineality spaces of convex cones
A matrix version of the Steinitz lemma
The Steinitz lemma, a classic from 1913, states that , a
sequence of vectors in with , can be rearranged so that
every partial sum of the rearranged sequence has norm at most .
In the matrix version is a matrix with entries
with . It is proved in \cite{OPW} that there
is a rearrangement of row of (for every ) such that the sum of the
entries in the first columns of the rearranged matrix has norm at most
(for every ). We improve this bound to
Curves in R^d intersecting every hyperplane at most d+1 times
By a curve in R^d we mean a continuous map gamma:I -> R^d, where I is a
closed interval. We call a curve gamma in R^d at most k crossing if it
intersects every hyperplane at most k times (counted with multiplicity). The at
most d crossing curves in R^d are often called convex curves and they form an
important class; a primary example is the moment curve
{(t,t^2,...,t^d):t\in[0,1]}. They are also closely related to Chebyshev
systems, which is a notion of considerable importance, e.g., in approximation
theory. We prove that for every d there is M=M(d) such that every at most d+1
crossing curve in R^d can be subdivided into at most M convex curves. As a
consequence, based on the work of Elias, Roldan, Safernova, and the second
author, we obtain an essentially tight lower bound for a geometric Ramsey-type
problem in R^d concerning order-type homogeneous sequences of points,
investigated in several previous papers.Comment: Corrected proof of Lemma 3.