TITLE: Pointwise infimum of affine functions is concave
QUESTION [19 upvotes]: 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.

REPLY [4 votes]: 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