Mean projections and finite packings of convex bodies (Q1335533)

The Sausage Conjecture of László Fejes Tóth (1975) says that the volume of the convex hull \(Q\) of a packing of \(n\) unit balls in \(\mathbb{E}^ d\), \(d \geq 5\), attains its minimum if the centers of the balls lie on a straight line. This has been proved recently for sufficiently large values of \(d\) by U. Betke, M. Henk and J. Wills. In contrast to that, the author shows that, for large \(n\), \(Q\) is almost spherical if the mean \(i\)-dimensional projection of \(Q\), \(1\leq i<d\), is minimized. (For \(i=d- 1\), this means the surface area of \(Q\).) Actually, the author proves the corresponding theorem for packings of \(n\) translates of an arbitrary convex body \(C\) by deriving a specific lower bound on the quotient of inradius and circumradius of \(Q\). Analogous results are obtained for lattice packings and in case the affine hull of the centers of the circumscribed ball of the copies of \(C\) is restricted to lower dimensions.