Affine Restriction estimates imply Affine Isoperimetric inequalities

One thing I absolutely love about harmonic analysis is that it really has something interesting to say about nearly every other field of Analysis. Today’s example is exactly of this kind: I will show how a Fourier Restriction estimate can say something about Affine Geometry. This was first noted by Carbery and Ziesler (see below for references).

1. Affine Isoperimetric Inequality

Recall the Affine Invariant Surface Measure that we have defined in a previous post. Given a hypersurface $\Sigma \subset \mathbb{R}^d$ sufficiently smooth to have a well-defined Gaussian curvature $\kappa_{\Sigma}(\xi)$ (where $\xi$ ranges over $\Sigma$) and with surface measure denoted by $d\sigma_{\Sigma}$, we can define the Affine Invariant Surface measure as the weighted surface measure

$\displaystyle d\Omega_{\Sigma}(\xi) := |\kappa_{\Sigma}(\xi)|^{1/(d+1)} \, d\sigma_{\Sigma}(\xi);$

this measure has the property of being invariant under the action of $SL(\mathbb{R}^d)$ – hence the name. Here invariant means that if $\varphi$ is an equi-affine map (thus volume preserving) then

$\displaystyle \Omega_{\varphi(\Sigma)}(\varphi(E)) = \Omega_{\Sigma}(E)$

for any measurable $E \subseteq \Sigma$.
The Affine Invariant Surface measure can be used to formulate a very interesting result in Affine Differential Geometry – an inequality of isoperimetric type. Let $K \subset \mathbb{R}^d$ be a convex body – say, centred in the origin and symmetric with respect to it, i.e. $K = - K$. We denote by $\partial K$ the boundary of the convex body $K$ and we can assume for the sake of the argument that $\partial K$ is sufficiently smooth – for example, piecewise $C^2$-regular, so that the Gaussian curvature is defined at every point except maybe a $\mathcal{H}^{d-1}$-null set. Then the Affine Isoperimetric Inequality says that (with $\Omega = \Omega_{\partial K}$)

$\displaystyle \boxed{ \Omega(\partial K)^{d+1} \lesssim |K|^{d-1}. } \ \ \ \ \ \ \ (\dagger)$

Notice that the inequality is invariant with respect to the action of $SL(\mathbb{R}^d)$ indeed – thanks to the fact that $d\Omega$ is. Observe also the curious fact that this inequality goes in the opposite direction with respect to the better known Isoperimetric Inequality of Geometric Measure Theory! Indeed, the latter says (let’s say in the usual $\mathbb{R}^d$) that (a power of) the volume of a measurable set is controlled by (a power of) the perimeter of the set; more precisely, for any measurable $E \subset \mathbb{R}^d$

$\displaystyle |E|^{d-1} \lesssim P(E)^d,$

where $P(E)$ denotes the perimeter1 of $E$ – in case $E = K$ a symmetric convex body as above we would have $P(K) = \sigma(\partial K)$. But in the affine context the “affine perimeter” is $\Omega(\partial K)$ and is controlled by the volume instead of viceversa. This makes perfect sense: if $K$ is taken to be a cube $Q$ then $\kappa_{\partial Q} = 0$ and so the “affine perimeter” cannot control anything. Notice also that the power of the perimeter is $d$ for the standard isoperimetric inequality and it is instead $d+1$ for the affine isoperimetric inequality. Informally speaking, this is related to the fact that the affine perimeter is measuring curvature too instead of just area.
So, the inequality should actually be called something like “Affine anti-Isoperimetric inequality” to better reflect this, but I don’t get to choose the names.

The inequality above is formulated for convex bodies since those are the most relevant objects for Affine Geometry. However, below we will see that Harmonic Analysis provides a sweeping generalisation of the inequality to arbitrary hypersurfaces that are not necessarily boundaries of convex bodies. Before showing this generalisation, we need to introduce Affine Fourier restriction estimates, which we do in the next section.