The Loomis-Whitney inequality is a non-sharp inequality that gives a upper-bound on the volume of a compact euclidean manifold in 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.
Let be the standard basis vectors in and let:
be the orthogonal projection onto the hyperplane perpendicular to .
Theorem: Loomis-Whitney inequality
Suppose that are non-negative. Then:
As an application of , let be a compact manifold and let:
so implies that .
From , we obtain:
Now, for the special case of a compact manifold with given surface area , this implies that we have:
We may easily prove in the case of 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