# Analytic center

### From Glossary

Given the set, , which we assume
is non-empty and compact, such that is
concave on , with a
non-empty strict interior,
its *analytic center* is the (unique) solution to the
*maximum entropy* problem:

Note that the analytic center depends on how the set is defined -- i.e., the nature of , rather than just the set, itself. For example, consider the analytic center of the box, . One form is to have functions as: . In this case, the analytic center is for all , where is the solution to:

Since , the analytical center is what we usually think of as the center of the box. However, the upper bounds could be defined by for all , where (so is concave). This changes the functional definition, but not the set -- it's still the unit box. The analytic center is skewed towards the corner because the defining mathematical program is:

The solution is , so the analytic center approaches as .