# 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.

## 2. Affine Fourier Restriction estimates

It was noted from quite early in the history of Fourier restriction that the (allowed) range of boundedness of the Fourier restriction operator for hypersurface $\Sigma$, given by $f \mapsto \widehat{f}\,\big|_{\Sigma}$, depends on how well-curved $\Sigma$ is. In particular, the Knapp example2 (see page 6ff. of this survey on Fourier restriction by Tao) shows that a necessary condition for the restriction inequality

$\displaystyle \| \widehat{f}\,\big|_{\Sigma}\|_{L^q(\Sigma,d\sigma)} \lesssim \|f\|_{L^p(\mathbb{R}^d)}$

to hold is that

$\displaystyle \frac{d-1}{q} + \frac{d+1}{p} \geq d+1; \ \ \ \ \ \ \ (\text{Kn})$

we will say that a pair of exponents $p,q$ for which the above is an equality are on the critical line.
Suppose however that the Gaussian curvature $\kappa_{\Sigma}$ is identically vanishing on (a portion of) $\Sigma$ – for example, you can imagine taking the hypersurface to be a truncated cone. In this case the adapted Knapp example shows that the necessary condition becomes as follows: if $k$ principal curvatures of $\Sigma$ (the eigenvalues of the shape operator…) are non-vanishing across $\Sigma$ then the necessary condition becomes

$\displaystyle \frac{k}{q} + \frac{k+2}{p} \geq k+2$

(you can check that for $k=d-1$ one recovers condition (Kn)). You can verify (by e.g. plotting the region on a Riesz diagram with axes $\frac{1}{p}, \frac{1}{q}$) that the range allowed by this condition is smaller than the one allowed by (Kn) when $k < d-1$, which shows that "worse curvature" means "worse Fourier restriction" if you allow me this rough expression. There are other ways in which the curvature can worsen, for example if the hypersurface has points that are flatter than a non-degenerate quadratic hypersurface but not totally flat – say, the contact along any tangent line at that point is of order higher than 2.
All of this is reflected in practice in the fact that typically Fourier restriction estimates for hypersurfaces are proven under the assumption that the Gaussian curvature is bounded from below (at least in absolute value), with the constant depending implicitely on this lowerbound – and necessarily blowing up as it approaches zero, by the above conditions (that is, if the estimates reach all the way to the critical line). When the hypersurface has no lowerbound on the Gaussian curvature, things start to get messy.

Flatness of varying degrees is therefore going to be a problem for Fourier restriction, at least in the sense that the number of situations to consider increases considerably. However, clever people have thought of a way to bypass this issue and obtain general estimates that are at least partially independent of the particular surface under consideration.
The original seed goes back to Sjölin’s work on Fourier restriction to curves in $\mathbb{R}^2$, in which he showed uniform (or universal, if you prefer) restriction theorems for a large class of curves: the idea was to replace the arclength measure with a special weighted arclength measure. This weighted measure is the Affine Invariant Arclength Measure, the 1-dimensional cousin of the affine invariant surface measure $d\Omega$ above. I will say more about that measure in the future, but for the moment I avoid defining it entirely and observe that in $\mathbb{R}^2$ curves and surfaces are the same thing, so it coincides with the affine invariant surface measure. In particular, if $t \mapsto \boldsymbol{\gamma}(t)$ is a parametrisation of a curve, its curvature at $\boldsymbol{\gamma}(t)$ is given by $\kappa(t) = \det\begin{bmatrix} \boldsymbol{\gamma}'(t) & \boldsymbol{\gamma}''(t) \end{bmatrix} \|\boldsymbol{\gamma}'(t)\|^{-3}$ and its “surface” measure is the arclength measure $\| \boldsymbol{\gamma}'(t)\| \,dt$; therefore in this case we have

$\displaystyle d\Omega_{\boldsymbol{\gamma}}(t) = \begin{vmatrix} \boldsymbol{\gamma}'(t) & \boldsymbol{\gamma}''(t) \end{vmatrix}^{1/3} \,dt.$

What Sjölin proved is that for $p,q$ lying on the critical line – that is, $3q = p'$ – and such that $1 < p < 4/3$ one has the Affine Invariant Fourier restriction estimate

$\displaystyle \big\| \widehat{f} \, \big|_{\boldsymbol{\gamma}} \big\|_{L^q(d\Omega_{\boldsymbol{\gamma}})} \lesssim_{p} \|f\|_{L^p(\mathbb{R}^2)};$

notice that this estimate with these parameters is only affine invariant for $p,q$ on the critical line, and if one wanted affine invariant estimates for $p,q$ inside the boundedness region one would have to change the exponent in the density of $d\Omega_{\boldsymbol{\gamma}}(t)$ to something that depends on $p,q$ themselves – in particular, one would need to use measure $\begin{vmatrix} \boldsymbol{\gamma}'(t) & \boldsymbol{\gamma}''(t) \end{vmatrix}^{q/{p'}} \,dt$.

Inspired by Sjölin’s work, Carbery and Ziesler looked at hypersurfaces (possibly with “problematic” curvature) and considered affine invariant Fourier restriction estimates for these. The analogue of Sjölin’s measure is the affine invariant surface measure $d\Omega_{\Sigma}$ that we have introduced above and thus what they looked for is estimates of the form

$\displaystyle \big\| \widehat{f} \, \big|_{\Sigma} \big\|_{L^q(d\Omega_{\Sigma})} \lesssim_{p} \|f\|_{L^p(\mathbb{R}^d)},$

where $p,q$ are on the critical line; notice that again this inequality is invariant under affine transformations (but if you go out of the critical line you need to change the exponent in the density of $d\Omega_{\Sigma}$ to retain affine invariance, as before).

An estimate of this form would be uniform (or universal) across a class of hypersurfaces (possibly only constrained by basic regularity assumptions) if the constant could be taken to be independent of the particular hypersurface – all the dependence would be subsumed in the measure $d\Omega_{\Sigma}$. They considered whether it’s possible to establish these uniform estimates at least in the well-understood Stein-Tomas range3 but ran into a counterexample when considering the broadest class of hypersurfaces (simply those of $C^2$ regularity). Their counterexample consists of a highly oscillating surface (see Theorem 1.3 in their paper) and this shows that one cannot have uniformity across a large class of hypersurfaces without some sort of bounded multiplicity assumption. Since in dimensions higher than 2 the Stein-Tomas estimates are more-or-less equivalent to decay estimates for the Fourier transform of the characteristic function of the weighted hypersurface (in this case, that would be $\widehat{d\Omega_{\Sigma}}(x)$), they went on to identify some decay estimates4 that, if true, imply affine invariant Stein-Tomas estimates. Furthermore, they identify classes of radial hypersurfaces in all dimensions that satisfy such estimates.

I highly recommend seeing the introduction in Carbery and Ziesler’s paper and the references therein for more information about what I briefly and roughly discussed above.

Before I close this section, let me mention that D. Oberlin has shown a weak-type Fourier restriction estimate that is uniform over all hypersurfaces in $\mathbb{R}^d$ of bounded multiplicity (see paper for a definition). His proof is extremely short and relies on choosing a particular exponent $p$ for which the corresponding restriction estimate on the critical line can be restated in a geometric-combinatorial form (thus devoid of cancellation phenomena). More precisely, he has shown that for a $C^2$ hypersurface $\Sigma$ one has

$\displaystyle \big\| \widehat{f} \, \big|_{\Sigma} \big\|_{L^{4(d-1)/(d+1), \infty} (d\Omega_{\Sigma})} \lesssim_{m(\Sigma)} \|f\|_{L^{4/3}},$

where $m(\Sigma)$ denotes the multiplicity5 of $\Sigma$. The key point is that with $p = 4/3$ one has $p' = 4 = 2 \times 2$ and this allows one to multilinearise the adjoint extension estimate. I will present D. Oberlin’s proof in the future.

## 3. Affine Fourier Restriction implies Affine Isoperimetric Inequalities

In their paper, Carbery and Ziesler also observed that affine invariant Fourier restriction estimates are strong enough to show affine isoperimetric inequalities – which is the main thing I wanted to present today. More precisely, we will show that any affine invariant Fourier restriction estimate along the critical line implies the affine isoperimetric inequality for any measurable subset of the hypersurface – a sensibly stronger result than the typical affine isoperimetric inequality for convex bodies!

Given a set $E \subset \mathbb{R}^d$ we denote by $\mathrm{ch}(E)$ its convex hull, that is

$\displaystyle \mathrm{ch}(E) := \{ (1-\theta)x + \theta y \,: \, x,y \in E, \theta \in [0,1] \},$

and by $\overline{\mathrm{ch}}(E)$ the closure of its convex hull. Then the Extended Affine Isoperimetric Inequality that we are going to show today can be stated as follows:

Definition: Let $\Sigma_0$ be a hypersurface. $\Sigma_0$ is said to satisfy the Extended Affine Isoperimetric Inequality if for every measurable subset $\Sigma \subseteq \Sigma_0$ it holds that (with $\Omega = \Omega_{\Sigma_0}$)

$\displaystyle \Omega (\Sigma)^{d+1} \lesssim |\overline{\mathrm{ch}}(\Sigma)|^{d-1}.$

Notice that while $\overline{\mathrm{ch}}(\Sigma)$ is a convex body by definition, $\Sigma$ is NOT its boundary in general! For example, if $\Sigma$ is hyperbolic it is going to be contained in the interior of $\overline{\mathrm{ch}}(\Sigma)$. Thus the Extended Affine Isoperimetric inequality is much stronger than the Affine Isoperimetric Inequality ($\dagger$).
We are going to show the following.

Proposition 1: If an affine invariant Fourier restriction inequality on the critical line holds for the hypersurface $\Sigma_0$ , then $\Sigma_0$ satisfies the Extended Affine Isoperimetric inequality.

In full, if $p,q$ are such that $\frac{d-1}{q}+\frac{d+1}{p} = d+1$ and we have

$\displaystyle \big\| \widehat{f} \, \big|_{\Sigma_0} \big\|_{L^q(d\Omega_{\Sigma_0})} \lesssim \|f\|_{L^p(\mathbb{R}^d)}$

for every $f \in L^p(\mathbb{R}^d)$, then for every measurable $\Sigma \subseteq \Sigma_0$ we have

$\displaystyle \boxed{ \Omega (\Sigma)^{d+1} \lesssim |\overline{\mathrm{ch}}(\Sigma)|^{d-1}. } \ \ \ \ \ \ \ \ (\ddagger)$

By the way, to recover ($\dagger$) from the Proposition just take $\Sigma_0 = \partial K$.

In this final section we are going to prove this Proposition, but before we do that we need a small preliminary fact.

### 3.1. Reverse Blaschke-Santaló inequality

Another important result in Affine Geometry is the Blaschke-Santaló inequality. Let $K \subset \mathbb{R}^d$ be a convex symmetric body as before; its convex dual (or polar body) is the convex set

$\displaystyle K^\ast := \{ x \in \mathbb{R}^d \, : \, \forall \xi \in K, \, |x \cdot \xi| \leq 1 \}.$

The Blaschke-Santaló inequality says that $K, K^\ast$ are dual in the following sense: for every $K$ convex symmetric body centred in the origin it holds that

$\displaystyle |K| |K^\ast| \leq |B(0,1)|^2.$

We know that the constant is sharp and equality is obtained only when $K$ is any affine image of the unit ball $B(0,1)$ (an ellipsoid). Surprisingly, if one is not concerned with the best constant (which for affine geometers is very important, but bear with me) the inequality can be proved quickly using just Plancherel’s theorem. Indeed, observe the following about the Fourier transform of $K$: if $x \in \frac{1}{100} K^{\ast}$ we have $|x \cdot \xi | \leq 1/100$ for every $\xi \in K$, and therefore for such an $x$ we have

\displaystyle \begin{aligned} |\widehat{\mathbf{1}_K}(x)| = & \Big| \int_{K} e^{-2\pi i x \cdot \xi} \,d\xi \Big| \\ \geq & \Big| \int_{K} \cos(2\pi x \cdot \xi) \,d\xi \Big| \\ \gtrsim & \int_{K} 1 \,d\xi = |K|. \end{aligned}

Using this lowerbound we get by Plancherel

$\displaystyle |K| |K^\ast|^{1/2} \lesssim \| \widehat{\mathbf{1}_K} \|_{L^2} = \| \mathbf{1}_K\|_{L^2} = |K|^{1/2},$

from which it is immediate that $|K||K^\ast| \lesssim 1$ (with a bad constant). This calculation will be useful below. Even more surprisingly, one can also give a completely Fourier analytic proof of the Blaschke-Santaló inequality with the optimal constant by using less trivial means. See this paper of Bianchi and Kelly for the nice proof. Alternatively, see this post of Văn Hoàng for a shorter account of the proof.

However, interestingly some sort of “affine uncertainty principle” also holds, in the form of a reverse Blaschke-Santaló inequality. That is, for $K \subset \mathbb{R}^d$ it holds for some dimensional constant that

$\displaystyle |K||K^\ast| \gtrsim_d 1. \ \ \ \ \ \ \ (\text{rB-S})$

For this direction of the inequality we don’t know what the best constant is! We have a conjecture that says that the best constant should be the one given by taking $K$ to be a cube6, but we don’t know how to prove it. The reverse inequality above was first proven by Mahler in 1939, with a constant that is $1/d!$ of the conjectured one. Such bounds with bad constants are not hard to prove, and so it is fine for our purposes in this post to consider the inequality above with a bad dimensional constant elementary. Regarding the best constant however, the first huge step forward -and currently still the best result in this regard, up to the value of $C$ below- was obtained by Bourgain (of course) and Milman, who proved that for some $C > 0$

$\displaystyle |K||K^\ast| \geq C^{-d}|Q||Q^\ast|,$

with $Q$ the unit cube. Their paper is justly celebrated as a great achievement. It is Charlie Fefferman’s favourite paper of Bourgain! A rough sketch of the proof can be found in the Proceedings of Thiele’s Summer School of 2019.

### 3.2. Proof of Proposition 1

After much setting up, we are finally ready to prove the Proposition.
Let $\Sigma_0$ be a (sufficiently smooth) hypersurface and let $\Omega = \Omega_{\Sigma_0}$ denote its Affine Invariant Surface Measure. We assume that the Affine Fourier Restriction estimate

$\displaystyle \| \widehat{f}\,\big|_{\Sigma_0}\|_{L^q(d\Omega)} \lesssim \|f\|_{L^p(\mathbb{R}^d)}$

holds for exponents $p,q$ on the critical line, that is $p,q$ are such that $\frac{d-1}{q} + \frac{d+1}{p} = d+1$. Observe that this latter condition can be rewritten as $\frac{q}{p'} = \frac{d-1}{d+1}$.
By duality, the adjoint (weighted) Fourier extension estimate

$\displaystyle \| \widehat{g \, d\Omega} \|_{L^{p'}(\mathbb{R}^d)} \lesssim \|g\|_{L^{q'}(d\Omega)}$

is also true: indeed

\displaystyle \begin{aligned} \| \widehat{g \, d\Omega} \|_{L^{p'}(\mathbb{R}^d)} = & \sup_{\|f\|_{L^p(\mathbb{R}^d)} \leq 1} \int_{\mathbb{R}^d} f(x) \widehat{g \, d\Omega}(x) \,dx \\ = & \sup_{\|f\|_{L^p(\mathbb{R}^d)} \leq 1} \int_{\Sigma_0} \widehat{f}(\xi) g(\xi) \, d\Omega(\xi) \\ \leq & \sup_{\|f\|_{L^p(\mathbb{R}^d)} \leq 1} \big\| \widehat{f}\big|_{\Sigma_0} \big\|_{L^{q}(d\Omega)} \|g\|_{L^{q'}(d\Omega)} \\ \lesssim & \sup_{\|f\|_{L^p(\mathbb{R}^d)} \leq 1} \|f\|_{L^p(\mathbb{R}^d)} \|g\|_{L^{q'}(d\Omega)} = \|g\|_{L^{q'}(d\Omega)}. \end{aligned}

So, for a measurable subset $\Sigma \subseteq \Sigma_0$ take $g \equiv \mathbf{1}_{\Sigma}$: $\|\mathbf{1}_{\Sigma}\|_{L^{q'}(d\Omega)}$, the RHS of the affine Fourier extension inequality, is precisely $\Omega(\Sigma)^{1/{q'}}$. For the LHS, let $K := \overline{\mathrm{ch}}(\Sigma)$ and let $K^\ast$ denote its convex dual as before; then, by the same calculation as above (Section 3.1.), if $x \in \frac{1}{100} K^\ast$ we see that since $\Sigma \subset \overline{\mathrm{ch}}(\Sigma)$ we have

\displaystyle \begin{aligned} |\widehat{\mathbf{1}_{\Sigma} \, d\Omega}(x)| = & \Big| \int_{\Sigma} e^{-2\pi i \xi \cdot x} |\kappa(\xi)|^{1/(d+1)} \, d\sigma(\xi) \Big| \\ \geq & \Big| \int_{\Sigma} \cos(2\pi \xi \cdot x) |\kappa(\xi)|^{1/(d+1)} \, d\sigma(\xi) \Big| \\ \gtrsim & \int_{\Sigma} |\kappa(\xi)|^{1/(d+1)} \, d\sigma(\xi) = \Omega(\Sigma). \end{aligned}

Using this pointwise bound we can lowerbound

$\displaystyle \| \widehat{\mathbf{1}_{\Sigma} \, d\Omega} \|_{L^{p'}(\mathbb{R}^d)} \gtrsim \Omega(\Sigma) |K^\ast|^{1/{p'}},$

and therefore we have

$\displaystyle \Omega(\Sigma) |K^\ast|^{1/{p'}} \lesssim \| \widehat{\mathbf{1}_{\Sigma} \, d\Omega} \|_{L^{p'}(\mathbb{R}^d)} \lesssim \|\mathbf{1}_{\Sigma}\|_{L^{q'}(d\Omega)} = \Omega(\Sigma)^{1/{q'}}.$

Rearranging the terms we have shown

$\displaystyle \Omega(\Sigma) \lesssim |K^\ast|^{-q/{p'}}.$

On the RHS we want $|K|$ instead, so we appeal to the reverse Blaschke-Santaló inequality (rB-S) (and here is where we actually need to have chosen $K$ to be the convex hull of $\Sigma$) to have

$\displaystyle \Omega(\Sigma) \lesssim_d |K|^{q/{p'}};$

as we have already observed, $\frac{q}{p'} = \frac{d-1}{d+1}$, so if we raise both sides to the power $d+1$ we have the desired extended affine isoperimetric inequality ($\ddagger$). $\square$

Footnotes:
1: There are different ways of defining the perimeter of an arbitrary measurable set. The most common and versatile one is to define $P(E)$ to be the total variation of the function $\mathbf{1}_E$, that is

$\displaystyle P(E):= \sup \Big\{ \int_{\mathbb{R}^d} \mathbf{1}_E \, \mathrm{div} \boldsymbol{\phi} \,dx \, : \, \boldsymbol{\phi} \in C^1_c(\mathbb{R}^d;\mathbb{R}^d), \|\boldsymbol{\phi}\|_{L^\infty} \leq 1 \Big\},$

a definition that doesn’t make sense the first time you see it until you realise that by Stoke’s theorem if you replace $\mathbf{1}_E$ with a differentiable function $f$ then the integral above becomes equal to $- \int_{\mathbb{R}^d} \langle \boldsymbol{\phi}, \nabla f \rangle \,dx$, and taking the supremum of this would then give $\int_{\mathbb{R}^d} \|\nabla f\| \,dx$, a more easily recognisable variation.
[go back]
2: Essentially, the Knapp example is obtained by testing the Fourier extension inequality (dual to the given restriction inequality) against the characteristic function of a small cap on the surface; the conditions on the exponents are then obtained by letting the size of the cap go to zero. [go back]
3: The Stein-Tomas range has endpoint $q=2$ along the critical line, thus $\frac{d-1}{2} + \frac{d+1}{p} = d+1$ and hence $p = \frac{2d+2}{d+3}$. [go back]
4: We record here the estimate for the interested reader. Let $\Sigma$ be given by a graph $\mathbb{R}^{d-1} \ni \boldsymbol{t} \mapsto (\boldsymbol{t}, \phi(\boldsymbol{t}))$; then since $\kappa_{\Sigma}(\boldsymbol{t}) = \frac{\det(\nabla^2 \phi(\boldsymbol{t}))}{(1 + \|\nabla \phi(\boldsymbol{t})\|^2)^{(d+1)/2}}$ and $d\sigma_{\Sigma} = (1 + \|\nabla \phi(\boldsymbol{t})\|^2)^{1/2} \,d\boldsymbol{t}$ we have $d\Omega_{\Sigma} = |\det(\nabla^2 \phi(\boldsymbol{t}))|^{1/(d+1)}\, d\boldsymbol{t}$. Then the desired estimate is (with $x =(x',x_d) \in \mathbb{R}^{d-1} \times \mathbb{R}$)

$\displaystyle \Big| \int e^{- i(x' \cdot \boldsymbol{t} + x_d \phi(\boldsymbol{t}))} |\det(\nabla^2 \phi(\boldsymbol{t}))|^{\beta + i \alpha} \, d\boldsymbol{t} \Big| \lesssim \frac{(1 + |\alpha|)^N}{ |x_d|^{\beta(d-1)}}$

for some $\beta \in [0,1/2]$, some $N > 0$ and for every $\alpha \in \mathbb{R}$. $\beta = 1/2$ gives the classical Stein-Tomas endpoint exponent. [go back]
5: For completeness, the multiplicity of $\Sigma$ is defined as follows. Assuming that $\Sigma$ is given by the graph $\mathbb{R}^{d-1} \ni \boldsymbol{t} \mapsto \Phi(\boldsymbol{t})=(\boldsymbol{t}, \phi(\boldsymbol{t}))$, let $\Psi(\boldsymbol{t}_1, \ldots, \boldsymbol{t}_d) := (\Phi(\boldsymbol{t}_d)-\Phi(\boldsymbol{t}_1), \ldots, \Phi(\boldsymbol{t}_d) - \Phi(\boldsymbol{t}_{d-1}))$; then

$\displaystyle m(\Sigma) := \sup_{y \in (\mathbb{R}^d)^{d-1}} \# \Psi^{-1}(\{y\}).$

[go back]
6: Interestingly, there are lots of other (conjectured) minimisers for the reverse Blaschke-Santaló inequality. Besides the cube, convex bodies that give the same constant are octahedra (defined in dimension $d$ to be the unit ball with respect to $\ell^1$ norm), products of (lower dimensional) cubes and octahedra, convex duals of these sets, products of these convex duals, etc. [go back]