Stability theorems for two measures of symmetry (Q1580752)

The paper proposes two new measures of symmetry of a convex body \(K\) in Euclidean \(n\)-space \(\mathbb{R}^n\). The first measure is \(\eta(K)= \sup\{h_K (u)/h_K (-u):u\in S^{n-1}\}\), where \(h_K(u)\) denotes the support function of \(K\) with respect to the centroid of \(K\). The second measure is \(\theta(K)=\text{sup\{vol}(K_u)/\text{vol}(K_{-u}):u\in S^{n-1}\}\) with \(K_u=K\cap E^+(u)\), where \(E^+(u)\) denotes the half-space whose boundary hyperplane contains the centroid of \(K\) and has \(u\) as an interior normal vector. Cones appear to be the most asymmetric convex bodies for both the measures. Here a cone means the convex hull of the union of a convex compact subset of a hyperplane and of a point out of this hyperplane. The author establishes stability estimates for both measures which say about the deviation of a convex body from a cone. For instance, the estimate for the measure \(\eta\) says that for every \(\varepsilon\geq 0\) such that \(\eta(K)\geq n-\varepsilon\) there exists a cone \(C\) such that \(\text{vol} (K\cup C)-\text{vol}(K\cap C)\leq {4\over n+1}\text{vol} (K)\varepsilon\).