Context. I'm reading the manuscrip "Nonlinear Error Bounds via a Change of Function" by Dominique Azé and Jean-Noël Corvellec (J Optim Theory Appl 2016), and I'm having a hard time understanding a certain condition in their Propositino 3.2.

Let $X=(X,d)$ be a complete metric space with connected balls. Given a continuous function $f:X \to \mathbb R$ and a point $x \in X$, the strong slope of $f$ at $x$, denoted $|\partial|^- f(x)$, is defined by

$$ |\partial |^- f(x) := \limsup_{y \to x}\frac{(f(x)-f(y))_+}{d(x,y)}. $$

Strong slopes are very important in analysing gradient-flows in general spaces (e.g metric spaces, manifolds, probability measure spaces, etc.).

Question. What are sufficient conditions on $f$ to ensure that $|\partial |^- f(x) = |\partial|^- (-f)(x)$ for all $x \in X$?

N.B. I'm fine with restricting the problem to the case where $X$ is a Banach space and $f$ is convex, in which case we have

$$ |\partial|^- f(x) = \sup_{y \ne x}\frac{f(x)-f(y)}{\|x-y\|}, $$

whenever $x$ is not a minimum point of $f$. In particular, I have in mind functions of the form $f(x) := \max \{\langle a_i,x\rangle \mid i=1,\ldots,n\}$, for some bounded linear operators $a_1,\ldots,a_n \in X^\star$.

Context. I'm reading the manuscrip "Nonlinear Error Bounds via a Change of Function" by Dominique Azé and Jean-Noël Corvellec (J Optim Theory Appl 2016), and I'm having a hard time understanding a certain condition in their Propositino 3.2.

Let $X=(X,d)$ be a complete metric space with connected balls. Given a continuous function $f:X \to \mathbb R$ and a point $x \in X$, the strong slope of $f$ at $x$, denoted $|\partial|^- f(x)$, is defined by

$$ |\partial |^- f(x) := \limsup_{y \to x}\frac{(f(x)-f(y))_+}{d(x,y)}. $$

Strong slopes are very important in analysing gradient-flows in general spaces (e.g metric spaces, manifolds, probability measure spaces, etc.).

Question. What are sufficient conditions on $f$ to ensure that $|\partial |^- f(x) = |\partial|^- (-f)(x)$ for all $x \in X$?

N.B. I'm fine with restricting the problem to the case where $X$ is a Banach space with topological dual $X^\star$ and $f$ is convex, in which case we have that if $x$ is not a minimum point of $f$, then

$$ |\partial|^- f(x) = \sup_{y \ne x}\frac{f(x)-f(y)}{\|x-y\|} = \|\partial f(x)\|_\star = \{\|x^\star\|_\star \mid x^\star \in \partial f(x)\}, $$

where $\partial f(x) \subseteq X^\star$ is the subdifferential of $f$ at $x$. In particular, I have in mind functions of the form $f(x) := \max \{\langle a_i,x\rangle \mid i=1,\ldots,n\}$, for some bounded linear operators $a_1,\ldots,a_n \in X^\star$.