The author considers the isoperimetric problem of minimizing perimeter for given volume in a strictly convex bounded domain \(\Omega\subset\mathbb{R}^{n+1}\) with \(C^2\) boundary. Searching for conditions which guarantee that an isoperimetric region in \(\Omega\) is convex he proves the following main result (Theorem 1.1): if \(\Omega\) is rotationally symmetric about some line, then any isoperimetric region in \(\Omega\) is convex. Moreover – by constructing a special rotationally symmetric convex set \(\Omega\) in \(\mathbb{R}^3\) (Fig. 1) he shows that the theorem above is independent of the “great circle condition” which means that a largest open ball \(B\) in \(\Omega\) has a great circle contained in \(\Omega\). In 1997, E. Stredulinsky and William P. Ziemer proved that this great circle condition is also sufficient (but obviously not necessary) for any isoperimetric region in \(\Omega\) to be convex. If \(\Omega\) satisfies a great circle condition then isoperimetric regions \(E\) in \(\Omega\) for large volumes are unique and nested: in precise terms one has \(E_1\subset E_2\) whenever \(|B|\leq|E_1|< |E_2|\), \(B\) the union of all largest balls in \(\Omega\). It is well-known that one cannot expect nestedness when \(\Omega\) is non-convex. The last section of this note is devoted to the construction of an example of a strictly convex body with non-nested minimizers.