A variational formula for the Lutwak affine surface areas j of convex bodies in Rn is established when 1 ≤ j ≤ n − 1. By using introduced new ellipsoids associated with projection functions of convex bodies, we prove a sharp isoperimetric inequality for j , which opens up a new passage to attack the longstanding Lutwak conjecture in convex geometry.

The paper deals with a variational approach of the subRiemannian geometry from the point of view of Hamilton-Jacobi and Hamiltonian formalism. We present a discussion of geodesics from the point of view of both formalisms, and prove that the normal geodesics are locally length-minimizing horizontal curves.

We construct many new non-liftable three-dimensional Calabi–Yau spaces in positive characteristic. The technique relies on lifting a nodal model to a smooth rigid Calabi–Yau space over some number field as introduced by one of us jointily with D. van Straten.

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