# Functional Covering Numbers

## TL;DR

This paper introduces and analyzes covering and separation numbers for functions, establishing their properties, exact equalities for certain classes, and geometric inequalities, including strong M-position results for geometric log-concave functions.

## Contribution

It defines new function-based covering and separation numbers, explores their properties, and extends geometric inequalities to these concepts, providing new insights into log-concave functions.

## Key findings

- Exact equality of separation and covering numbers for some function classes
- Analogues of geometric inequalities for covering numbers
- Strong versions of M-positions for geometric log-concave functions

## Abstract

We define covering and separation numbers for functions. We investigate their properties, and show that for some classes of functions there is exact equality of separation and covering. We provide analogues for various geometric inequalities on covering numbers, such as volume bounds, bounds connected with Hadwiger's conjecture, and inequalities about M-positions for geometric log-concave functions. In particular, we obtain strong versions of M-positions for geometric log-concave functions.