# Viterbo's conjecture for certain Hamiltonians in classical mechanics

## TL;DR

This paper investigates Viterbo's conjecture in the context of specific classical mechanical Hamiltonians, establishing the conjecture for certain convex 2-homogeneous cases and discussing unresolved instances.

## Contribution

It provides new proofs of Viterbo's conjecture for particular convex Hamiltonians of classical mechanics, expanding the understanding of the conjecture's validity.

## Key findings

- Viterbo's conjecture holds for certain convex 2-homogeneous Hamiltonians.
- The paper identifies specific cases where the conjecture is proven.
- Open cases of the conjecture are discussed for future research.

## Abstract

We study some particular cases of Viterbo's conjecture relating volumes of convex bodies and actions of closed characteristics on their boundaries, focusing on the case of a Hamiltonian of classical mechanical type, splitting into summands depending on the coordinates and the momentum separately. We manage to establish the conjecture for sublevel sets of convex $2$-homogeneous Hamiltonians of this kind in several particular cases. We also discuss open cases of this conjecture.