Property of convex, two times differentiatable functions, concerning gradients This is my first post here. I am looking for some hint, how to prove the following equation:
$$\nabla f(x)^\top x + \nabla f(-x)^\top(-x) \ge 0.$$
I have no clue, how this property is called. We just did some properties of convex functions, in particular if they are 1x or 2x differentiable. Does anyone know the name of this property or in which book to find it?
Thanks a lot
Cheers
 A: This is a specification of a more general property of $\nabla f$, known as monotonicity. In particular, a set-valued map $T : \Bbb{R}^n \rightrightarrows \Bbb{R}^n$ is monotone if, for all $x, y \in \Bbb{R}^n$, and $x^* \in T(x)$, $y^* \in T(y)$, we have
$$\langle x - y, x^* - y^* \rangle \ge 0.$$
Here, $\langle \cdot, \cdot \rangle$ refers to the dot product, i.e. $\langle p, q \rangle = p^\top q$. I write it with alternate notation, because this concept does generalise beyond $\Bbb{R}^n$.$^1$ In this case, given $\nabla f$ is single-valued, this turns into,
$$\langle x - y, \nabla f(x) - \nabla f (y) \rangle = (\nabla f(x) - \nabla f(y))^\top(x - y) \ge 0,$$
for all $x, y \in \Bbb{R}^n$. What you have written is the special case where $y = -x$:
$$0 \le (\nabla f(x) - \nabla f(-x))^\top(x - (-x)) = 2\nabla f(x)^\top x + 2\nabla f(-x)^\top(-x),$$
and dividing by $2$ yields the result.
Yes, it is true that gradient maps of convex functions are monotone.$^2$ As requested, here’s a hint how to prove it: recall that, if $f$ is convex, the tangent plane at $y \in \Bbb{R}^n$ minorises $f$. That is, for any $y \in \Bbb{R}^n$, $\nabla f(y)^\top(x - y) + f(y) \le f(x)$ for all $x$. What if you swap the roles of $x$ and $y$?

$^1$ Indeed, the same definition works in Hilbert spaces, or if $X$ and $Y$ are paired Banach spaces (in particular, if $Y$ is the dual of $X$), then the same definition works for $T : X \rightrightarrows Y$ instead. But, you don’t have to concern yourself with this!
$^2$ In fact, we can even do without the differentiability! So long as $f$ is a proper, convex, lower-semicontinuous map, the subderivative map $\partial f$ has the monotonicity property. This is a set-valued generalisation of the derivative used (primarily) for convex functions. When $f$ is convex and differentiable at $x$, $\partial f(x) = \{\nabla f(x)\}$, so it does generalise the derivative.
