Date: Sunday, 9 Adar II 5765 (March 20, '05)

Speaker: Evelyn Magazanik (Hebrew University, Jerusalem)

Title: "Visibility theory and its generalizations"

Abstract:
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Let $x$ and $y$ be two points in a set $S \subset \R^d$. We say that these
points see each other via $S$ if the segment joining them lies within $S$.
This definition of visibility leads to the central concepts of star and
kernel, whose properties have been widely investigated by many
mathematicians.

Lately, the concepts of polygonal visibility (two points see each other if
there exists a polygonal path of at most $n$ edges connecting them) and
staircase visibility (the edges of the polygonal path are parallel to the
coordinate axes, and monotone in each coordinate) have been defined for
planar sets. It is natural, then, to define the concepts of polygonal and
staircase stars and kernels, and ask which of their properties still hold.

We will define ordinary stars and kernels and the concepts of high and low
visibility, and exhibit many families of special subsets ("crowns") whose
intersection is the convex kernel of $S$. We will also investigate
polygonal stars and kernels, including the concepts of diameter and radius
(defined in a paper by Micha Perles and myself), and generalize the
concept of a crown. If time permits, we'll define staircase visibility and
generalize many of the previous results.

(joint work with M. A. Perles)