# Jerusalem Mathematics Colloquium

Thursday, 11 June 1998, 4:00 pm

Mathematics Bldg., lecture hall 2

## Dr. Menachem Kojman (Ben-Gurion University)

## "Degrees of non-convexity"

** Abstract: **
Non-convex sets in linear spaces are divided to sets
which are finite unions of convex sets, sets which are countable unions of
convex set and all others.

A hierarchy of length * omega*_{1 } of non-convexity
of closed sets in separable Banach spaces will be presented, which refines
the class of sets which are countable unions of convex sets.

First the problem of covering by convex sets is reduced to computing the
chromatic number of a certain hypergraph. Using topology, an ordinal rank
of a set is defined and it is seen that a bounded rank is sufficient for
countable chromatic number. Finally, in Polish linear spaces, the
sufficient condition is also necessary. Thus, to every closed set in a
Polish linear space there corresponds an ordinal * rho * so that the
set is a countable union of convex sets if and only if * rho * is
countable. This ordinal is the "non-convexity degree" of the set.

A similar hierarchy was previously known for closed sets in the plane.

Coffee, Cookies at the faculty lounge at 3:30.

List of talks, 1997-98