This research is about questions of stability in geometric inequalities for convex bodies. Progress has recently been made by others in this area and three problems which arise in this context will be studied. The in- and circumradius of convex sets appear in the inequalities which establish stability or in their proofs, and additional work on inequalities of this type may lead to the resolution of a conjecture which originated in algebraic geometry. Bonnesen-style inequalities are involved in some cases and the investigator has studied these extensively.
