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

Abstract

For convex bodies D in R" the deviation d from spherical shape is estimated from above in terms of the (dimensionless) isoperimetric deficiency A of D as follows: d < /(A) (for A sufficiently small). Here / is an explicit elementary function vanishing continuously at 0. The estimate is sharp as regards the order of magnitude of /. The dimensions n = 2 and 3 present anomalies as to the form of /. 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 R" in the sense that, as D varies, -0 for A - 0 . The proof of the estimate d < /(A) 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.