Suppose that $f_{1}(x) ... f_{m}(x)$ are piece wise linear convex functions from $\mathbb{R}^n$ to $\mathbb{R} $ and let given $f(x) = \sum_{i=0}^{m} f_{i}(x)$. Show that $f(x)$ is also piece wise linear and convex functions.
Using the definition that a piece wise function $ = max_{j=1...m}(c_{j}^{T}x + d_{j})$ $c_{j}$is is a vector and $d_{j}$ is a scalar
How would you prove this?
Thanks