Pointwise infimum of affine functions is concave So I was just starting on convex optimization and was having a slightly hard time visualizing the lagrangian being always concave because it is the pointwise infimum  of a family of affine functions. 
Can anyone help explain this? I've googled extensively but most places just state this without elaboration or examples.
Thanks.
 A: See picture below for some intuition on why that may be (bold line is the pointwise infimum of the four affine functions). Similarly, you can conclude that the

Pointwise supremum of the set of affine functions is convex


A: Daniel Fischer gave a transparent explanation in terms of epigraphs $\{(x,y): y\ge f(x)\}$:

A function is convex if and only if its epigraph is convex, and the epigraph of a pointwise supremum is the intersection of the epigraphs. [Hence,] the pointwise supremum of convex functions is convex. 

One can similarly argue from concavity,  using the sets $\{(x,y): y\le f(x)\}$: this set is convex if and only if $f$ is concave. Taking infimum of functions results in taking the intersection of such sets.
A: The point-wise maximum of a set of convex functions is convex.
$$g(x) = \max_i f_i(x) $$
The corresponding epigraph $$\{(x,t) | t > \max_i f_i(x)\}$$ is convex which could also be visualized as the insertion of a family of convex spaces, $$\cap_i \{(x ,t)| t > f_i(x)\}$$
The lagrangian could be rewritten as the negative of the supremum,
$$lagrangian(x) = \min_i f_i(x) = - \max_i -f_i(x)$$
