On the Aleksandrov--Fenchel inequality and the stability of the sphere (Q1261138)

Let \(M\subset E^ d\) be a convex body and let \(0 \leq\varepsilon<1\). If the mean curvature \(H_ 1\) of \(K\) is bounded by \(1-\varepsilon \leq H_ 1 \leq 1+\varepsilon\) then there is a ball \(B_ K \subset E^ d\) satisfying \[ \delta_ 2(K,B_ K) \leq \sqrt \varepsilon V(K) V(B^ d)^{-1} \left( {2^{3d-4}(d-1)^{2d-3)} \over d+1} \right)^{1/2} \] where \(\delta_ 2(K,L)\) denotes the \(L_ 2\)-distance of convex bodies \(K,L \subset E^ d\), given by \(\delta_ 2 (K,L)= \biggl( \omega(S^{d- 1})^{-1} \int_{S^{d-1}}\bigl( h_ K(u)-h_ L(u) \bigr)^ 2d \omega(u) \biggr)^{1/2}\) \(h_ K(u)\), \(h_ L(u)\) being the respective support functions and \(B_ K\) means the Steiner ball of \(K\) (the ball with the same mean width and Steiner point as \(K)\).