**Diclaimer.** I'm not sure this is the right venue for this question, but I'll give it a try

Definition [Strong / metric slope].Given a complete metric space $(M,d)$ and a function $f:M \to (-\infty,+\infty]$, define its strong slope at a point $x \in M$, denote $\partial |f|(x)$, by$$ |\partial|f(x) := \begin{cases}0,&\mbox{ if }f\text{ attains a local minimum at }x,\\ \limsup_{y \to x}\frac{f(x)-f(y)}{d(x,y)},&\mbox{else.}\end{cases} $$

This concept crops up in gradient-flow literature for metric spaces, probability spaces, etc. For more details on this concept, see this monograph and the references therein.

Now, consider the unit ball $\mathbb B_m := \{x \mid \|x\| \le 1\}$ in $\mathbb R^m$, seen as a metric space equipped with euclidean distance.

## Question

Fix $n\ge 1$ points $a_1,\ldots,a_n \in \mathbb R^m$ and consider the convex function $f:\mathbb B_m \to \mathbb R$ defined by $f(x) := \max \{a_i^\top x \mid i = 1,\ldots,n\}$.

How to go about computing $|\partial|f(x)$ ?

**Note.** Ultimately, I'm only interested in uniform lower-bound on $\partial |f|(x)$ for $\|x\| < 1$, i.e finding $\alpha > 0$ such that $\inf_{x \in \mathbb B_m^\circ}\partial |f|(x) \ge \alpha$.

Any useful hints will be very much appreciated.

## Observations

- I know how to solve the problem in case the domain of $f$ is replaced with the entire flat space $\mathbb R^m$. Indeed, in this case, I can prove that $|\partial|f(x) \ge \gamma := \min_{q \in \Delta_{n-1}}\|A^\top q\|$, where $A$ is the $n \times m$ matrix with $i$th row equal to $a_i$.
- In the spherical case (the setup of my problem), even the completely linear scenario where $n=1$, i.e $f(x) \equiv a_1^\top x$, is already not clear to me.
- In the case of Banach spaces (of which $\mathbb B_m$ is not!), there are known uniform lower-bounds in terms of Hadamard derivatives.