I read by M. Fabian, P. Habala etc., p. 338, lemma 7.13, which is the following.
Let $C>0$ and $f$ be a $C$-Lipschitz convex function defined on a nonempty open and convex subset $U$ of a Banach space $X$. Let $x_0\in U$ and $\epsilon>0$ be such that $B(x_0,\epsilon)\subset U$. Then, $\partial f(B(x_0,\epsilon)\subset \partial_{2C\epsilon}f(x_0)$.
The definition of $\epsilon$-subdifferential of a proper function $f$ at $x_0$ (denoted by $\partial_\epsilon f(x_0)$) is given by the set of dual elements $x^*$ such that $f(x)\ge f(x_0)+<x^*,x-x_0>-\epsilon \forall x\in X$ and we omit the $\epsilon$ when $\epsilon=0$.
In the proof of lemma it first fix $x\in B(x_0,\epsilon), x^*\in \partial f(x)$ and $y \in B(x_0,\epsilon)$, then
$f(y)\ge f(x)+<x^*,y-x>$
$=f(x_0)+<x^*,y-x_0>+(f(x)-f(x_0))+<x^*,x_0-x>$
$\ge f(x_0)+<x^*,y-x_0>-2C\Vert x-x_0\Vert\ge f(x_0)+<x^*,y-x_0>-2C\epsilon$
and by the convexity the conclustion follows.
I understand all these things, but I think it does not needs convexity. At the first time we chose $y \in B(x_0,\epsilon)$, but all the inequalities hold for $y\in U$ so we do not use convexity argument. Am I missed something?