### Abstract:

In this study we investigate a measure of the so-called
joint separation
of clusters
from both the experimental and the theoretical points of view. In the experimental part,
we measure the
joint separation
of a finite set of planes
F
of cardinality
n
∏
2
in term
of the
minmax
angle of vectors on the planes belonging to a given set.
Minmax
angle
is defined as the largest angle
μ
such that, if
n
vectors are chosen, one each on the
plane, then at least two of the vectors have an angle of at least
μ
between them. Since
obtaining
minmax
angle for arbitrary finite sets of plane in
R
3
seems hard, we restrict to
planes bounding the faces of a regular polyhedron. An algorithm based on Hill Climbing
and Lingo model is used in the search for
minmax
angle of regular polyhedron. In the
theoretical part, we study the generation of
minmax
angle as a notion of
joint separation
.
Given a set of points of various colors on the line, an interval is called
color-spanning
if
it contains at least one point of each color. We present an efficient algorithm to solve a
problem of finding the minimum
color spanning
interval. Furthermore a semi-dynamic
and a fully- dynamic algorithm to maintain such interval are proposed.