Girth Functions of Convex Bodies: Some Remarks

1. Some remarks on the girth functions of convex bodies Paolo Gronchi Dipartimento di Matematica e Applicazioni per l’Architettura Università degli Studi di Firenze (joint work with Stefano Campi) Cortona 2007

2. u u 3-dimensional case higher dimensions ________________ ________________ u u What is the girth function? K gK(u) = mean width (K|u) gK(u) = perimeter (K|u) /  Cortona 2007

3. unit ball segment [u,-u] convex area measure of order 1 First property of the girth function gK(u) is a support function (when extended homogeneously) Cortona 2007

4. gK(u) is the support function of Π1K , the projection body of order 1 of K Π1K is a zonoid Cortona 2007

5. All Minkowski sums of disks are projections bodies of order 1. K = segment parallel to u Π1K = disk in u Not all zonoids are projection bodies of order 1. Cortona 2007

6. This suggests that hΠK(u) is not independent of its values on u. 1 In higher dimensions In dimension 3, such an ellipsoid cannot be the sum of disks. It is easy to prove that, for any K, Cortona 2007

7. This suggests that hΠK(u) is not independent of its values on u. 1 Integrating last inequality we find Cortona 2007

8. is equivalent to the existence of a box P such that Cortona 2007

9. u False negative number! The existence of a cylinder C such that is equivalent to the statements as a function of vu is a support function. Cortona 2007

10. Cortona 2007

11. Cortona 2007