Let $f: \mathbb{R}^n \mapsto \mathbb{R}$ a convex a continuously differentiable function.

What is the relation between the *component Lipschitz constants*, i.e. the smallest constants $L_i$ such that for all $x \in \mathbb{R}^n$ and $t \in \mathbb{R}$:
\begin{equation}
|[\nabla f(x+te_i)]_i-[\nabla f(x)]_i| \leq L_i|t|,
\end{equation}
and the regular Lipschitz constant, i.e. the smallest constant $L$ such that for all $d \in \mathbb{R}^n$:
\begin{equation}
\|\nabla f(x+d)-\nabla f(x)\|_2 \leq L\|d\|_2?
\end{equation}

It is mentioned it this paper that $1 \leq \frac{L}{L_{\text{max}}} \leq n$ where $L_{\text{max}} = \max_{i=1..n}L_i$, which comes from the "relationships between norm and trace of a symmetric matrix" (p.12). However I do not see where this result comes from, or which "symmetric matrix" the author refers to.