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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A cute combinatorial result of Santal√≥

There is a nice result due to Santaló that says that if a (finite) collection of axis-parallel rectangles is such that any small subcollection is aligned, then the whole collection is aligned. This is kind of surprising at first, because the condition only says that there is a line, but this line might be different for any choice of subcollection. The precise statement is as follows:

Theorem. Let \mathcal{R} be a collection of rectangles with sides parallel to the axes (possibly intersecting). If for every choice of 6 rectangles of \mathcal{R} there exists a line intersecting all 6 of them, then there exists a line intersecting all rectangles of \mathcal{R} at once.

To be precise, I should clarify that by line intersection it is meant intersection with the interior of the rectangle – so a line touching only the boundary is not allowed. The number 6 doesn’t have any special esoteric meaning here, to the best of my understanding – it just makes the argument work.
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Dimension of projected sets and Fourier restriction

Pdf version here: link.

I had a nice discussion with Tuomas after the very nice analysis seminar he gave for the harmonic analysis working group a while ago – he talked about the behaviour of Hausdorff dimension under projection operators and later we discussed the connection with Fourier restriction theory. Turns out there are points of contact but the results one gets are partial, and there are some a priori obstacles.

What follows is an account of the discussion. I will summarize his talk first.

1. Summary of the talk

1.1. Projections in {\mathbb{R}^2}

The problem of interest here is to determine whether there is any drop in the Hausdorff dimension of fractal sets when you project them on a lower dimensional vector space, and if so what can be said about the set of these “bad” projections. This is a very hard problem in general, so one has to start with low dimensions first. In {\mathbb{R}^2} the projections are associated to the points in {\mathbb{S}^1}, namely for {e\in\mathbb{S}^1} one has {\pi_e (x) = (x\cdot e)e}, and so for a given compact set {K} of Hausdorff dimension {0\leq \dim K \leq 1} one asks what can be said about the set of projections for which the dimension is smaller, i.e. {\dim \pi(K) < \dim K}. For {s \leq \dim K}, define the set of directions

\displaystyle E_s (K):= \{e \in \mathbb{S}^1 \,:\, \dim \pi_e (K) < s\}.

We refer to it as to the set of exceptional directions (of parameter {s}). One preliminary result is Marstrand’s theorem:

Theorem 1 (Marstrand) For any compact {K} in {\mathbb{R}^2} s.t. {s<\dim K <1}, one has

\displaystyle |E_s (K)| = 0.

In other words, the dimension is conserved for a.e. direction. The proof of the theorem relies on a characterization of dimension in terms of energy:

Theorem 2 (Frostman’s lemma) For {K} compact in {\mathbb{R}^d}, it is {s<\dim K} if and only if there exists a finite positive Borel measure {\mu} supported in {K} such that

\displaystyle I_s(\mu):= \int_{K}{\int_{K}{\frac{d\mu(x)\,d\mu(y)}{|x-y|^s}}}<\infty.

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Riemannian geometry I

I had many courses in my undergraduate studies, but for one reason or another never got to see some riemannian geometry. Which is really a shame, because I always wanted to learn some general relativity. So, here I am, studying riemann geometry for the first time. Since there’s lots of things in it, what follows is really just a summary, primarily for my own understanding.

Let {M} be a smooth manifold of dimension {n}, {TM = \sqcup_{p\in M}{T_p M}} his tangent bundle, {T^\ast M} the cotangent bundle, and let {\mathcal{T}(M)} denote the space of smooth sections of {TM} (i.e. the smooth vector fields on {M}). Analogously, {\mathcal{T}^{k}_{l}(M)} will denote the smooth sections of the tensor bundle {T^{k}_{l}M} – that is, the smooth tensor fields [1]. Tensors fields are multilinear with respect to {C^{\infty}(M)}. For example, {\mathcal{T}M = \mathcal{T}_{1}(M)}, while {\mathcal{T}^{1}(M)} are the {1}-forms. We also impose {\mathcal{T}^{0}(M)=C^{\infty}(M)}.

1. Riemannian metrics

A Riemannian metric on {M} is a {2}-tensor field (in {\mathcal{T}^2 (M)}) – given in local coordinates by {g_{ij}dx^i \otimes dx^j} – which is:

  1. symmetric: {g(X,Y) = g(Y,X)} for every {X, Y \in \mathcal{T}(M)};
  2. positive definite: {g(X,X)>0} for every {X\neq 0}.

It induces an inner product on every tangent space. Continue reading