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Let be a subset of a real or complex vector space
. The
is said to be convex if for all
and all
the point
is in
. In other words, every point on the line segment
connecting
and
is in
. The empty set is considered as convex.
If the set does not contain all the line segments, it is called concave or non-convex.
If is a convex set then for any
and
such that
, then the point
is also in
.
Convex hull of a set of a complex or real vector space
is the smallest convex set containing
. Since the intersection of any collection of convex sets is itself convex, the convex hull of
is the intersection of all convex sets containing
.
A set of a complex or real vector space is called star convex if there exists an
such that all line segments from
to any point
are contained in
. A convex set is always star convex but the converse is not true in general.
There are numerous geometry algorithms for computing of convex hulls in the plane. For the most common of these algorithms visit .
The convexity notion may be generalizes to other type of spaces or structures in such a way that the following properties are fulfilled:
Cite this web-page as:
Štefan Porubský: Convex sets.