Definition
Convex set
A set S (subset of /equation015.png)
: /equation.png)
) is a convex set, if and only if for every /equation005.png)
every convex combination of /equation016.png)
are also in the set.
Or: for all /equation006.png)
, /equation001.png)
Strictly convex set
A subset /equation004.png)
is strictly convex, if for all , /equation007.png)
and all /equation008.png)
/equation002.png)
Where /equation003.png)
is the interior of S.
Properties of convex set
Let A and B are convex subsets of /equation009.png)
:
(1) /equation010.png)
is convex
(2) /equation011.png)
is convex
(3) The union of two convex sets need not be convex.
(4) The half-spaces /equation012.png)
and the hyperplane /equation013.png)
are convex for all /equation014.png)