In this paper we consider the following analog of Bezout inequality for mixed volumes: V (P1, . . . , Pr,∆ )Vn(∆) r−1 ≤ r ∏ i=1 V (Pi,∆ ) for 2 ≤ r ≤ n. We show that the above inequality is true when ∆ is an n -dimensional simplex and P1, . . . , Pr are convex bodies in R . We conjecture that if the above inequality is true for all convex bodies P1, . . . , Pr , then ∆ must be an n -dimensional...

We prove that if a convex body has absolutely continuous surface area measure, whose density is sufficiently close to the constant, then the sequence {ΠK} of convex bodies converges to the ball with respect to the Banach-Mazur distance, asm→∞. Here, Π denotes the projection body operator. Our result allows us to show that the ellipsoid is a local solution to the conjectured inequality of Petty ...

It is shown that for any sufficiently regular even Minkowski valuation Φ which homogeneous and intertwines rigid motions, there exists a neighborhood of the unit ball, where balls are only solutions to fixed-point problem Φ2K=αK. This significantly generalizes results by Ivaki projection bodies suggests, via Lutwak–Schneider class reduction technique, new approach Petty's conjectured inequality.

In this paper, the concept of the classical f -divergence (for a pair of measures) is extended to the mixed f divergence (for multiple pairs of measures). The mixed f -divergence provides a way to measure the difference between multiple pairs of (probability) measures. Properties for the mixed f -divergence are established, such as permutation invariance and symmetry in distributions. An Alexan...