Only some remarks not fitting into a comment, but intended as a comment: what you describe (and explain more precisely in your comment) will give you something which I'd call the minimal radius, since you allow your planes to intersect the convex set. For 'diameter' I'd restrict to planes not intersecting the interior of the convex set. That is what I then would actually call diameter of the set. Of course these two should simply differ by a factor of 2. This (the minimal radius) is, for example, the geometric quantity you need if you want to estimate what time it takes to boil an egg, cause you need to calculate the amount of time it takes till every part of the egg is above a certain temperature ;-)
(What you want to do is a minimax search, that is you are looking for critical points of a certain functional which are neither global minima nor maxima, which is, in general, usually not completely trivial and a topic considered in Nonlinear functional analyis. Look, e.g., at Deimlings book on Nonlinear FA. For your specific problem this may be a bit too much firing power, but depending on your mathematical skills maybe you'll find it to be of interest. If only for terminology. I do not know about algorithms for this kind of problem, though I'm sure there are some. Note though, that you do not seem to be looking at sets with smooth boundary, which may imply additional complications. This is just a gut feeling of me, though.)