# Affine Invariant Surface Measure

In this short post I want to introduce an instance of certain objects that will be the subject of a few more posts. This particular object arises naturally in Affine Differential Geometry and turned out to have a relevant rôle in Harmonic Analysis too (in both Fourier restriction and in the theory of Radon transforms).

## 1. Affine Invariant measures

Affine Differential Geometry is the study of (differential-)geometric properties that are invariant with respect to $SL(\mathbb{R}^d)$. A very interesting object arising in Affine Geometry is the notion of an Affine Invariant Measure. Sticking to examples rather than theory (since the theory is still quite underdeveloped!), consider a hypersurface $\Sigma \subset \mathbb{R}^{d}$ sufficiently smooth to have well-defined Gaussian curvature, which we denote by $\kappa$ (a function on $\Sigma$). If we let $d\sigma$ denote the surface measure on $\Sigma$ (induced from the Lebesgue measure on the ambient space $\mathbb{R}^d$ for example, or by taking directly $d\sigma = d\mathcal{H}^{d-1}\big|_{\Sigma}$, the restriction of the $(d-1)$-dimensional Hausdorff measure to the hypersurface) then this crafty little object is called Affine Invariant Surface Measure and is given by

$\displaystyle d\Omega(\xi) = |\kappa(\xi)|^{1/(d+1)} \,d\sigma(\xi).$

It was first introduced by Blaschke for $d=3$ (finding the reference seems impossible; it’s [B] in this paper, if you feel luckier) and by Leichtweiss for general $d$. The reason this measure is so interesting is that it is (equi)affine invariant in the sense that if $\varphi(\xi) = A \xi + \eta$ is an equi-affine transformation (thus with $A \in SL(\mathbb{R}^d)$ and so volume-preserving since $\det A = \pm 1$) then, using subscripts to distinguish the two surfaces, we have

$\displaystyle \boxed{ \Omega_{\varphi(\Sigma)}(\varphi(E)) = \Omega_{\Sigma}(E) } \ \ \ \ \ \ \ (1)$

for any measurable $E \subseteq \Sigma$. We remark the following fact: that seemingly mysterious power $\frac{1}{d+1}$ in the definition of $d\Omega$ is the only exponent for which the resulting measure is (equi)affine-invariant.