M.Sc Student | Michal Lenkiewicz |
---|---|

Subject | Intersecting Convex Sets |

Department | Department of Mathematics |

Supervisor | Professor Emeritus Katchalski Meir |

Full Thesis text |

This work is closely related to a well known theorem of Eduard Helly on convex sets in d-dimensional Euclidean space.

Helly's theorem
asserts that for a finite family of convex sets in R^{d}, the
intersection of all sets is nonempty, provided that the intersection of every
d or less sets is non empty.

Meir Katchalski extended results of Branko Grünbaum and proved that the intersection of the members of such a family is at least k-dimensional, provided that the intersection of every f(k,d)=min{d,2(d-k)} sets or less is at least k-dimensional.

Another closely related result by Katchalski for such families is that the intersection of the members of such a family contains a ray, provided that the intersection of every 2d or less sets contains a ray.

We extend this result and prove that for every k at most d, the intersection of the members of the family contains a k-dimensional convex cone, provided that the intersection of every f(k,d) sets or less contains a convex cone of dimension k. The constant f(k,d) is best possible.

The proof
utilizes properties of families of convex cones with apex zero in R^{d}
for which the intersection of the members of the family is {0}.

We prove that
for any such family of convex cones {A_{1},?,A_{d}} with
apex 0 in R^{d} and intersection equal to {0}, there are disjoint
subsets T_{1},?,T_{t} of T={1,?,d}, such that A(T\T_{i})
is a subspace of dimension |T_{i}|-1, where A(T\T_{i}) is the
intersection of all sets A_{j}
of the family with j in T\T_{i}.

This is related
to results of Katchalski on nondegenerate families of convex cones. A nonempty
family of convex cones with apex 0 in R^{d} is called a nondegenerate
family (N.D.F) if each member of the family is d-dimensional and the
intersection of any r members of the family is at least (d-r)-dimensional for
0<r<d and the intersection of all members of the family is the origin.

We also prove and use results which force the intersection of a family to be at least 1-dimensional.

A typical such
result is the following: For {A_{1},?,A_{d}} a finite family
of convex sets in R^{d}, if the intersection of all sets is nonempty
and the sum of the dimensions of the intersection of all sets but one is greater
than d(d) then the dimension of the intersection of all members of the family
is greater than 0. Moreover, if there are exactly d sets in the family then
the dimension of the intersection of all members of the family is equal to d.