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

4. where n(x) is the outward unit normal at x where σ K is the area measure of K where x is the segment joining x to the origin (d- 1 )-dimensional Hausdorff measure

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

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

7. A proof of Petty’s inequality. Ingredients: Mixed volumes. L p -centroid body. Minkowski’s first inequality: V 1 (K, L) d  V(K) d-1 V(L), and equality holds if and only if K and L are homothetic. L p -Busemann-Petty centroid inequality: V(Γ p K)  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 with equality if and only if K and L are homothetic ellipsoids, with L origin-symmetric. We have

9. minimized when L is a level set of h ΠK, 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