In projective geometry we know that ‘well-behaved’ curves such as conics can be viewed in two ways: either as a locus of points or an envelope of lines (the curve being determined by being tangent to every line in the envelope). One of the little gems of the new superficially dry course on ‘Variational Principles’ is a method to give us a way of switching between these two views algebraically (provided the situation is suitably nice).

Firstly, we suppose that $y=f(x)$ describes a locus of points. How might we parameterise the tangents in a way that doesn’t involve points at all? Tangents are lines of the form $y=mx-c$, so in some sense there are two co-ordinates $(m,c)$ that describe the tangents uniquely. If we are to obtain an equation in a similar form $c=f^*(m)$ we will clearly require that $f$ is strictly convex (else $g$ will fail to be well-defined), so assume this is the case.

We then seek the $c$ such that $y=mx-c$ is the unique tangent to the curve $y=f(x)$ with gradient $m$. Equating the two values for $y$ we get that $c = mx_0 - f(x_0)$ where $x_0$ is the unique real number such that $f'(x_0) = m$. Nothing mind-blowing so far.

But this becomes rather more interesting once we remember the supporting-line inequality for convex functions (a convex function is always bigger than any tangent with equality at the point of tangency). Writing this down at our point we get that $f(x) \geq mx - mx_0 + f(x_0)$. Rearranging, we get

$mx- f(x) \leq mx_0 - f(x_0)$ for all $x$.

So writing this in a slightly different way, we get the cute formula that $f^*(m) = \text{max}_x(mx-f(x))$.

And for an encore, it’s now obvious (by projective duality) that if we do this transformation twice we get back to where we started (the curve stays fixed, and our function goes from describing the locus to describing the envelope to describing the locus again), so we get the nice relation that $f^{**} = f$ for all strictly convex $f$.