Lutwak’s intrinsic-volume projection-body minimization conjecture

Determine whether, for every convex body K in Euclidean space and every integer 1 ≤ i ≤ n − 1, the ratio V_{i+1}(Π_i K) / V_{i+1}(K)^i is minimized by Euclidean balls, where Π_i denotes the projection body map of order i.

Background

The paper identifies sharp lower bounds for intrinsic volumes of projection bodies as a largely unresolved problem. For a convex body K in R^n, Πi K is the projection body of order i, and V{i+1} denotes the corresponding intrinsic volume. Lutwak conjectured that the ratio V_{i+1}(Πi K) / V{i+1}(K)^i attains its minimum for Euclidean balls for every order 1 ≤ i ≤ n − 1.

The conjecture is known for i = 1. In the case i = n − 1, affine invariance implies that ellipsoids are also minimizers. The paper further states that Petty’s related conjectured projection inequality would imply Lutwak’s conjecture, but that Petty’s conjecture had only recently been proved in certain special cases.