Abstract
In the quadratic Aleksandrov-Fenchel inequality for mixed volumes, stated as inequality (1) below, whereC 1, ...,C n-2 are smooth convex bodies, equality holds only if the convex bodiesK andL are homothetic. Under stronger regularity assumptions onC 1,...,C n-2, a stability estimate is proved, expressing thatK andL are close to homothetic if equality is satisfied approximately. This is applied to estimate explicitly the deviation of a closed convex hypersurface with mean curvature close to one from a unit sphere.
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