Loomis-Whitney inequality

The Loomis-Whitney inequality is a non-sharp inequality that gives a upper-bound on the volume of a compact euclidean manifold in $\mathbb{R}^3$ which will be used in my next blog post. It’s main advantage over the isoperimetric inequality, which gives the best possible bound, is that it’s much simpler to prove.

Def: Projections

Let $\{e_j \}$ be the standard basis vectors in $\mathbb{R}^n$ and let:

\begin{aligned}\pi_j : \mathbb{R}^n \rightarrow \mathbb{R}^{n-1} \end{aligned}

\begin{aligned}\pi_j : x \rightarrow (x_1,x_2, ..., \hat{x_j}, ...,x_n) \end{aligned}

be the orthogonal projection onto the hyperplane perpendicular to $e_j$.

Theorem: Loomis-Whitney inequality

Suppose that $F_1, ..., F_n \in L^{n-1}(\mathbb{R}^{n-1})$ are non-negative. Then:

$\int_{\mathbb{R}^n} F_1(\pi_1 x)...F_n(\pi_n x) dx \leq||F_1||_{n-1}...||F_n||_{n-1} \quad (*)$

As an application of $(*)$, let $E \subset \mathbb{R}^n$ be a compact manifold and let:

$F_j(u) = \chi_{\pi_j (E)}(u)$

so $x \in E$ implies that $F_j(\pi_j x)=1$.

From $(*)$, we obtain:

$|E| = \int \chi_E \leq \int \prod_{j=1}^{n}F_j(\pi_j x) dx \leq \prod_{j=1}^{n} ||F_j||_{n-1} \leq |\partial E|^{\frac{n}{n-1}}$

Now, for the special case of a compact manifold $M \subset\mathbb{R}^3$ with given surface area $S$, this implies that we have:

$Vol(M) \leq S^{\frac{3}{2}} \quad (**)$

We may easily prove $(*)$ in the case of $n=3$ by applying Cauchy-Schwarz twice. This derivation is left as a useful exercise for the reader.

Note 1: The above derivation is taken from the notes of Tony Carbery.

Note 2: The Loomis-Whitney inequality only dates back to 1949 and their original paper has a very clever derivation by a combinatorial method. It’s only two pages long and I definitely