Depth with respect to a family of convex sets

Understand

We propose a notion of depth with respect to a finite family $\mathcal{F}$ of convex sets in $\mathbb{R}^d$ which we call $\text{dep}_\mathcal{F}$.

• We begin showing that $\text{dep}_\mathcal{F}$ satisfies some expected properties for a measure of depth and that this definition is closely related to the notion of depth proposed by J.
• Tukey.
• We show that some properties of Tukey depth extend to $\text{dep}_\mathcal{F}$ and we point out some key differences.