We discuss the local differential geometry of convex affine spheres in R and of minimal Lagrangian surfaces in Hermitian symmetric spaces. In each case, there is a natural metric and cubic differential holomorphic with respect to the induced conformal structure: these data come from the Blaschke metric and Pick form for the affine spheres and from the induced metric and second fundamental form ...

A constructive algorithm to implement convex recursive deletion regions via two-layer perceptrons has been presented in a recent study. In the algorithm, the absolute values of the weights become larger and larger when the number of nested layers of a convex recursive deletion region increases. In addition, the absolute values of the weights are determined according to the complexity of the str...

Divergence functions play a central role in information geometry. Given a manifold M, a divergence function D is a smooth, nonnegative function on the product manifold M ×M that achieves its global minimum of zero (with semi-positive definite Hessian) at those points that form its diagonal submanifold ∆M ⊂ M ×M. In this Chapter, we review how such divergence functions induce i) a statistical st...

Properties of several sorts of lattices of convex subsets of Rn are examined. The lattice of convex sets containing the origin turns out, for n > 1, to satisfy a set of identities strictly between those of the lattice of all convex subsets of Rn and the lattice of all convex subsets of R. The lattices of arbitrary, of open bounded, and of compact convex sets in Rn all satisfy the same identitie...

We define a class of L-convex-concave subsets of RP , where L is a projective line in RP . These are sets whose sections by any plane containing L are convex and concavely depend on this plane. We prove a version of Arnold’s conjecture for these sets, namely we prove that each such set contains a line.

We prove some first order regularity estimates for a class of convex functions in Carnot-Carathéodory spaces, generated by Hörmander vector fields. Our approach relies on both the structure of metric balls induced by Hörmander vector fields and local upper estimates for the corresponding subharmonic functions.

We define a class of L-convex-concave subsets of RP 3 , where L is a projective line in RP 3. These are sets whose sections by any plane containing L are convex and concavely depend on this plane. We prove a version of Arnold hypothesis for these sets, namely we prove that each such set contains a line.

The total variation (TV) measure is a key concept in the field of variational image analysis. Introduced by Rudin, Osher and Fatemi in connection with image denoising, it also provides the basis for convex structure-texture decompositions of image signals, image inpainting, and for globally optimal binary image segmentation by convex functional minimization. Concerning vector-valued image data,...