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.

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