【凸优化-03】Convex Functions: Running average of convex functions

Let f(t) be a convex function and define g(t) to be the running average of f(t)

$$g(t)=\frac{1}{t}{\int }_0^{t}f(\tau )d\tau$$
Then g is convex.

This is easy (not exactly) enough to prove just by differentiating twice.

但是如果$f(x)$不可导的，好像用不了这个方法。

在 math.stackexchange 上看到有人想用convexity preserving operation证明：Alternate way to prove running average of a convex function is convex?

以及关于这个问题，想要一个几何方面的解释的：
Running average of a convex function is convex

（一）两次求导：

（二）一个不太懂的另一个方法？