A result of Favard states that among all compact convex sets in \(\mathbb R^2\) of a given perimeter and area the one with largest circumradius is a symmetric lens, i.e. an intersection of two discs of the same radius. V. A. Zalgaller [St. Petersbg. Math. J. 5, No. 1, 177–188 (1994; Zbl 0804.52005)] showed that if \(S_0\) and \(V_0\) satisfy the isoperimetric inequality (\(S_0^3 \geq 36\pi V_0^2\)) then, among all convex bodies in \(\mathbb R^3\) with surface area \(\leq S_0\) and volume \(\geq V_0\) the unique body having maximal diameter is a mean-curvature spindle-shaped body of surface area \(S_0\) and volume \(V_0\). A mean-curvature (resp. Gauss-curvature) spindle shaped body is the convex hull of a surface of revolution with constant mean (resp. Gauss) curvature and a line segment along the axis.
Instead of using area as the replacement for perimeter, the authors consider mean width. They show that if \(W_0\) and \(V_0\) satisfy the corresponding isoperimetric inequality (\(\pi W_0^3 \geq 6V_0\)) then among all convex bodies with mean width \(\leq W_0\) and volume \(\geq V_0\) the unique body having maximal diameter is a Gauss-curvature spindle shaped body of mean width \(W_0\) and volume \(V_0\). This body has maximal circumradius among solids of revolution and (as with Zalgaller’s result) it is conjectured to have maximal circumradius among all convex bodies.