# Closure condition

### From Glossary

The conditinos that the closure of a nonempty
strict interior be equal to the level set:
. Here is an example where the
closure condition fails:

Note that is continuous and quasiconvex. Its level set is , but the strict interior, is , so its closure is only . We lose the flat portion, .

This is important in the use of interior point methods and
in stability. Here
are some functions that satisfy the closure condition:

- explicitly quasiconvex
- positively homogeneous (all degrees positive or all negative)
- monotonic (all strictly increasing or all strictly decreasing)

When equality constraints are present, there are two forms of extension of the closure condition: to consider the relative strict interior, and to consider "feasible sequences". The first generally assumes is affine, so the closure condition becomes:

and

Then, the closure condition is that and are not empty, and

and