Viterbo's conjecture as a worm problem

TL;DR

This paper explores the connection between Viterbo's conjecture in symplectic geometry and Minkowski worm problems, linking it to systolic inequalities, Mahler's conjecture, and the Wetzel problem, with implications for algorithmic bounds.

Contribution

It establishes a novel relation between Viterbo's conjecture and Minkowski worm problems, enabling transfer to other geometric conjectures and proposing an algorithmic approach for bounds.

Findings

Relation between Viterbo's conjecture and Minkowski worm problems established

Connection to systolic Minkowski billiard inequalities and Mahler's conjecture

Proposed algorithmic approach for lower bounds on the Wetzel problem

Abstract

In this paper, we relate Viterbo's conjecture from symplectic geometry to Minkowski versions of worm problems which are inspired by the well-known Moser worm problem from geometry. For the special case of Lagrangian products this relation provides a connection to systolic Minkowski billiard inequalities and Mahler's conjecture from convex geometry. Moreover, we use the above relation in order to transfer Viterbo's conjecture to a conjecture for the longstanding open Wetzel problem which also can be expressed as a systolic Euclidean billiard inequality and for which we discuss an algorithmic approach in order to find a new lower bound. Finally, we point out that the above mentioned relation between Viterbo's conjecture and Minkowski worm problems has a structural similarity to the known relationship between Bellmann's lost-in-a-forest problem and the original Moser worm problem.