In this post, I will prove Dines’ TheoremFun fact: Lloyd L. Dines was the chair of the math department at Carnegie Institute of Technology in the 30s–40s, prior to it merging with the Mellon Institute of Industrial Research to form CMU. (a neat fact about the convexity of a quadratic image of ) as well as a generalization to . These facts will be useful when I eventually get around to writing a post about hidden convexity and the S-lemma.
My friend Kevin Shu recently wrote a blog post giving a (relatively) elementary proof of Brickman’s Theorem, which is very related to this theorem.
Theorem 1 (Dines (1941)). Let , be real symmetric matrices. Then the image of under the quadratic map , given by is convex.
This is pretty surprising in my opinion: There are obvious counterexamples if one attempts to generalize this result. For example if one allows inhomogeneity, then the following set is the graph of a parabola and is nonconvex. Similarly, if one were to allow an additional quadratic form , then is (isomorphic to) the set of positive semidefinite rank-one matrices in , a nonconvex set (plotted below).
Clearly, both homogeneity and the fact that we are working in will play roles in any proof of Dines’ Theorem.
Proof. Let , and for notational convenience, let
First, suppose and are collinear in . Then, as is a cone, i.e., for every and , we have that the interval is contained in .
Next, suppose and are not collinear. Then, there exists a such that (i.e., and are both bounded away from the origin in some direction ).
Now, let where we will fix the sign later. In words, parameterizes the line containing and . Note that when we have , and when we have . We will pick the sign so that is also bounded away from the origin in the direction for all : Note that In particular, by picking the sign correctly, the last term above is nonnegative for all . Consequently, for all . Finally, leveraging the geometry of and the fact that is a cone, we deduce that the interval is contained in . ◻
What breaks for three quadratic forms?
If we look over the above proof line by line to see what breaks when we go from two quadratic forms to three quadratic forms, we will see that not too much breaks. Obviously, something does break because of the counterexample we saw earlier, but in fact the entirety of the proof remains intact save the very last sentence.
Specifically, in the case that and are not collinear, we can still find a path between and contained in the image and bounded away from zero in the direction . Unfortunately, we may no longer conclude that the interval is contained in , however, as this path may loop, so that it is not contained in the same plane as and . See for example, the indicated points in the plotted figure above.
The power of
Curiously, the three quadratic forms generalization of Dines’ Theorem is true when we switch from to .
Theorem 2 (???I remember reading about this result in some paper a while ago but don’t remember the correct reference at the moment. Will fix later.). Let be Hermitian matrices. Then the image of under the quadratic map , given by is convex.
The main difference between and , in this context, is that instead of simply being allowed to pick between a sign for , we now have the freedom to pick for any . Or, perhaps more suggestively, we get to pick for both some sign and some . This extra freedom, the choice of , will allow us to force the path that we construct between and onto the plane containing and . The rest of the proof follows analogously to Dines’ Theorem.
Dines, L. L. 1941. “On the Mapping of Quadratic Forms.” Bull. Amer. Math. Soc. 47 (6): 494–98.