Characterizations of differentiability for h-convex functions in stratified groups

TL;DR

This paper characterizes first and second order differentiability of h-convex functions in stratified groups using h-subdifferentials, linking Aleksandrov's second order differentiability to horizontal gradient differentiability.

Contribution

It provides a new characterization of differentiability for h-convex functions in stratified groups via h-subdifferentials and horizontal gradients.

Findings

First and second order differentiability characterized using h-subdifferentials.

Equivalence established between Aleksandrov's second order differentiability and horizontal gradient differentiability.

Framework extends understanding of convexity in stratified groups.

Abstract

Using the notion of h-subdifferential, we characterize both first and second order differentiability of h-convex functions in stratified groups. We show that Aleksandrov's second order differentiability of h-convex functions is equivalent to a suitable differentiability of their horizontal gradient.