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

One thought on “Loomis-Whitney inequality

  1. Pingback: Derivation of isoperimetric inequality from ideal gas equation – Kepler Lounge

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