# Min-increasing trajectory

*This article defines a property that can be evaluated for a trajectory on the space of functions on a manifold*

## Definition

Let be a manifold and be a function , where:

- denotes the time parameter, and varies in
- denotes the spatial parameter, and varies in

In other words, is a trajectory (or path) in the space of all functions from to .

Then, is said to be **min-increasing** if the function:

is a monotone increasing function. (the function defined above is called the timewise-min function for ).

The corresponding notion is of a **max-decreasing trajectory** -- viz a trajectory where the maximum (or supremum) keeps decreasing.