A uniqueness result for mean section bodies (Q655354)

For \(1\leq k\leq d\), the \(k\)-th mean section body, \(M_k(K)\), of a convex body \(K\) in \(\mathbb{R}^d\), is the Minkowski sum of all its sections by \(k\)-dimensional flats. The authors show that, for any fixed \(1<k<d\), the body \(K\) is uniquely determined by the body \(M_k(K)\), assuming \(\dim K\geq d-k+2\). This result was previously known only for centrally symmetric bodies, or, in full generality, only for the case \(k=2\).