163 research outputs found
Lines pinning lines
A line g is a transversal to a family F of convex polytopes in 3-dimensional
space if it intersects every member of F. If, in addition, g is an isolated
point of the space of line transversals to F, we say that F is a pinning of g.
We show that any minimal pinning of a line by convex polytopes such that no
face of a polytope is coplanar with the line has size at most eight. If, in
addition, the polytopes are disjoint, then it has size at most six. We
completely characterize configurations of disjoint polytopes that form minimal
pinnings of a line.Comment: 27 pages, 10 figure
Geometric Permutations of Non-Overlapping Unit Balls Revisited
Given four congruent balls in that have disjoint
interior and admit a line that intersects them in the order , we show
that the distance between the centers of consecutive balls is smaller than the
distance between the centers of and . This allows us to give a new short
proof that interior-disjoint congruent balls admit at most three geometric
permutations, two if . We also make a conjecture that would imply that
such balls admit at most two geometric permutations, and show that if
the conjecture is false, then there is a counter-example of a highly degenerate
nature
Vertex adjacencies in the set covering polyhedron
We describe the adjacency of vertices of the (unbounded version of the) set
covering polyhedron, in a similar way to the description given by Chvatal for
the stable set polytope. We find a sufficient condition for adjacency, and
characterize it with similar conditions in the case where the underlying matrix
is row circular. We apply our findings to show a new infinite family of
minimally nonideal matrices.Comment: Minor revision, 22 pages, 3 figure
A necessary and sufficient condition for -transversals in
A -transversal to family of sets in is a -dimensional
affine subspace that intersects each set of the family. In 1957 Hadwiger
provided a necessary and sufficient condition for a family of pairwise
disjoint, planar convex sets to have a -transversal. After a series of three
papers among the authors Goodman, Pollack, and Wenger from 1988 to 1990,
Hadwiger's Theorem was extended to necessary and sufficient conditions for
-transversals to finite families of convex sets in with
no disjointness condition on the family of sets. However, no such conditions
for a finite family of convex sets in to have a -transversal
for has previously been proven or conjectured. We make progress in
this direction by providing necessary and sufficient conditions for a finite
family of convex sets in to have a -transversal
- …