(from Geometric Tomography , R.J.Gardner, 1995)

3. (from Geometric Tomography, R.J.Gardner, 1995)

4. (d-1)-dimensional Hausdorff measure where σK is the area measure of K where x is the segment joining x to the origin where n(x) is the outward unit normal at x (from Geometric Tomography, R.J.Gardner, 1995)

5. What about the volume of ΠK ? Petty’s conjecture: ellipsoids are minimizers Brannen’s conjecture: simplices are maximizers

6. * Petty’s inequality: ellipsoids are minimizers Zhang’s inequality: simplices are maximizers

7. A proof of Petty’s inequality. Ingredients: Mixed volumes. Minkowski’s first inequality: V1(K, L)d V(K)d-1V(L) , and equality holds if and only if K and L are homothetic. Lp-centroid body. Lp-Busemann-Petty centroid inequality: V(ΓpK)  V(K) , and equality holds if and only if K is a centered ellipsoid. Lutwak, Yang, Zhang (2000), Campi, G. (2002)

8. A proof of Petty’s inequality. Consider the functional We have with equality if and only if K and L are homothetic ellipsoids, with L origin-symmetric.

9. A proof of Petty’s inequality. minimized when L is a level set of hΠK , so when L is a dilatation of Π*K

10. Petty’s projection inequality Petty’s conjecture Lutwak’s inequality

11. An extension of Lutwak’s inequality