Central sections of centrally symmetric convex bodies (Q1097518)

Let K,K'\(\subset E\) d be two centrally symmetric bodies. The author considers the following famous problem: Let for each hyperplane \(H\ni 0\) be \[ Vol_{d-1}(K\cap H)< Vol_{d-1}(K'\cap H). \] Does then hold \(Vol_ d(K)< Vol_ d(K')?\) Larman and Rogers (1975) gave by probabilistic arguments a counterexample for \(d\geq 12\). From a recent result by K. Ball (1986) follows that \(K=C\) d (unit cube) and \(K'=rB\) d (the corresponding ball with suitable radius r) are counterexamples for \(d\geq d_ 0\). In this paper the author shows that some standard methods are not strong enough for counterexamples in the 3-dimensional case.