Stability in the isoperimetric problem for convex or nearly spherical domains in ${\bf R}\sp n$

For convex bodies $D$ in ${{\mathbf{R}}^n}$ the deviation $d$ from spherical shape is estimated from above in terms of the (dimensionless) isoperimetric deficiency $\Delta$ of $D$ as follows: $d \leq f(\Delta)$ (for $\Delta$ sufficiently small). Here $f$ is an explicit elementary function vanishing continuously at 0. The estimate is sharp as regards the order of magnitude of $f$. The dimensions $n = 2$ and $3$ present anomalies as to the form of $f$. In the planar case $n = 2$ the result is contained in an inequality due to T. Bonnesen. A qualitative consequence of the present result is that there is stability in the classical isoperimetric problem for convex bodies $D$ in ${{\mathbf{R}}^n}$ in the sense that, as $D$ varies, $d \to 0$ for $\Delta \to 0$. The proof of the estimate $d \leq f(\Delta)$ is based on a related estimate in the case of domains (not necessarily convex) that are supposed a priori to be nearly spherical in a certain sense.