Displacing Lagrangian toric fibers via probes

TL;DR

This paper investigates the geometric structure of monotone moment polytopes using probes, identifying conditions under which points are displaceable and relating these to the Ewald conjecture and Floer homology of Lagrangian fibers.

Contribution

It demonstrates that every rational polytope has a unique non-displaceable central point and characterizes displaceability of other points via the star Ewald condition, especially in low dimensions.

Findings

Every rational polytope has a unique interior integral point that is not displaceable.

In dimensions up to three, all monotone polytopes satisfy the star Ewald condition.

Displaceability of points correlates with the star Ewald condition, linking geometric and symplectic properties.

Abstract

This note studies the geometric structure of monotone moment polytopes (the duals of smooth Fano polytopes) using probes. The latter are line segments that enter the polytope at an interior point of a facet and whose direction is integrally transverse to this facet. A point inside the polytope is displaceable by a probe if it lies less than half way along it. Using a construction due to Fukaya-Oh-Ohta-Ono, we show that every rational polytope has a central point that is not displaceable by probes. In the monotone (or more generally, the reflexive) case, this central point is its unique interior integral point. In the monotone case, every other point is displaceable by probes if and only if the polytope satisfies the star Ewald condition. (This is a strong version of the Ewald conjecture concerning the integral symmetric points in the polytope.) Further, in dimensions up to and including three every monotone polytope is star Ewald. These results are closely related to the Fukaya-Oh-Ohta-Ono calculations of the Floer homology of the Lagrangian fibers of a toric symplectic manifold, and have applications to questions introduced by Entov-Polterovich about the displaceability of these fibers.