1,234 research outputs found
Topological transversals to a family of convex sets
Let be a family of compact convex sets in . We say
that has a \emph{topological -transversal of index }
(, ) if there are, homologically, as many transversal
-planes to as -planes containing a fixed -plane in
.
Clearly, if has a -transversal plane, then
has a topological -transversal of index for and . The converse is not true in general.
We prove that for a family of compact convex sets in
a topological -transversal of index implies an
ordinary -transversal. We use this result, together with the
multiplication formulas for Schubert cocycles, the Lusternik-Schnirelmann
category of the Grassmannian, and different versions of the colorful Helly
theorem by B\'ar\'any and Lov\'asz, to obtain some geometric consequences
Line transversals to disjoint balls
We prove that the set of directions of lines intersecting three disjoint
balls in in a given order is a strictly convex subset of . We then
generalize this result to disjoint balls in . As a consequence, we can
improve upon several old and new results on line transversals to disjoint balls
in arbitrary dimension, such as bounds on the number of connected components
and Helly-type theorems.Comment: 21 pages, includes figure
Codimension two and three Kneser Transversals
Let be integers with and let
be a finite set of points in . A -plane
transversal to the convex hulls of all -sets of is called Kneser
transversal. If in addition contains points of , then
is called complete Kneser transversal.In this paper, we present various
results on the existence of (complete) Kneser transversals for .
In order to do this, we introduce the notions of stability and instability for
(complete) Kneser transversals. We first give a stability result for
collections of points in with
and . We then present a description of
Kneser transversals of collections of points in
with for . We show that
either is a complete Kneser transversal or it contains
points and the remaining points of are matched in pairs in
such a way that intersects the corresponding closed segments determined by
them. The latter leads to new upper and lower bounds (in the case when and ) for defined as the maximum positive integer
such that every set of points (not necessarily in general position) in
admit a Kneser transversal.Finally, by using oriented matroid
machinery, we present some computational results (closely related to the
stability and unstability notions). We determine the existence of (complete)
Kneser transversals for each of the different order types of
configurations of points in
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
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