by Soren Larson

In this blog post I focus on interesting characteristics of convexity that we didn’t get to cover in class, borrowing problems and examples from Kreyszig’s “Introductory Functional Analysis with Applications” and Rockafellar’s famous “Convex Analysis” text. I’m particularly excited to discuss a theorem from Rockafellar’s text, as this text has been cited in virtually every analysis course lecture I’ve ever taken, though I’ve not gotten an opportunity yet to consider it. I suppose that the reader is familiar with basic notions of analysis.

I borrow this example from Problem 3 of Kreyszig’s Chapter 6.2. For normed space and finite dimensional subspace, and for a given we want to best approximate with an element . To do this, most sensible will be to find a basis for and approximate using a linear combination for scalars chosen to minimize the distance between and our linear combination approximation. So, putting

we look to find that minimizes . Before we proceed, we might like to be sure that is continuous in , so we can be sure that doesn’t `jump’ at a point e.g., of a possible minimum (e.g., is not well defined).

So, taking to be arbitrary fixed, writing and , and putting we see that whenever and so , by triangle inequality we have

But was arbitrary, so we’re done. So is continuous in , as desired.

Now we proceed to minimization of . Indeed, we recall that convexity gives us that any local minimum is also a global minimum, giving us a guarantee that any minimum we find is *the *minimum.

I claim that is convex in . That is, I plan to show for and , we have for fixed . To see this, noticing and again applying triangle inequality we have

as desired! This result is particularly useful, e.g., in linear regression for normed cost function!

I now change gears from functional analysis to convex analysis. Indeed, students of analysis often like to know if properties of functions are preserved when taken to the limit. In this way, it’d be particularly nice to know that \emph{convexity} is preserved in the limit. I first make this claim precise, and then I prove it.

Take to be a sequence of finite convex functions on open convex set of finite dimensional normed space. I claim that if pointwise (i.e., for and arbitrary fixed, we may find large enough such that whenever ), then is convex. I prove this claim.

Let be arbitrary, and take . Since pointwise, clearly we have

and

But since is convex, we have

and so

as desired. So, as we’d expect, convexity is preserved when taken to the limit. Rockafellar proves a more general form of this statement (in his Theorem 10.8), using weaker assumptions and more ideas from analysis. But since most of these ideas are outside the scope of this course, and the assumptions I use are reasonable or `practical’ ones, I leave it here.