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Let be a complex vector space.
A **set of matrices over** consists of subsets
for all
. Subsets
can be considered
as sets of matrices over by putting
for
. More
generally, if for some the corresponding set is not given a priori, put
this as the empty set.
For sets of matrices, we have the following notions of convexity:^{60}
A set of matrices over is called
**matrix convex** or a **matrix convex
set**
[Wit84b] if for all
and

,

and for all and
with
.

A set of matrices over is called
**absolutely matrix convex** [EW97a] if for all
and

,

and for all ,
and
with
,
.

A set of matrices over is called a
**matrix cone** [Pow74] if for all
and

,

and for all and
.

A set of matrices over is matrix convex if and only if all
matrix convex combinations of elements of are again in . is absolutely
matrix convex if and only if all
absolutely matrix convex combinations of elements of are again in .
Here, a
**matrix convex combination** of
, ..., (
) is a sum of the form
with matrices
such that
. An
**absolutely matrix convex combination** of , ..., is a sum
of the form
with matrices
and
such that
and
.

If is a topological vector space, topological terminology is to be considered at
all matrix levels:
For instance, a set of matrices over is called
**closed** if all are closed.

#### Footnotes

- ... convexity:
^{60}
- With the
notation
, a set of matrices over is simply a subset
of . The sets can be regained as
.
Then the definitions have the following form:

Let
. is called **matrix convex** if

is called **absolutely matrix convex** if

is called a **matrix cone** if

**Subsections**

** Next:** Examples
** Up:** Convexity
** Previous:** Convexity
** Contents**
** Index**
Prof. Gerd Wittstock
2001-01-07