Inductio Ex Machina

Just when I thought I was starting to get my head around the multitudinous uses of convexity in statistics I was thrown by the following definition:

A function f over the interval (a,b) is convex if, for all choices of {x,y,z} satisfying a < x < y < z < b the determinant

is non-negative.

After expanding the determinant and some algebraic twiddling I realised that this is just a very compact way of requiring that

which, after noticing that (z-y) + (y-x) = (z-x), of course is the more traditional way of saying a function is convex.

What’s neat about this determinant representation is that it extends nicely to what are known as k^th-order convex functions (ones whose derivatives up to order k are convex). Specifically, f is k-convex whenever satisfy and

While it is arguably less transparent than explicitly writing out all the convexity inequalities for each of the derivatives of f it certainly makes up for it with compactness.