How was Claim 5 in "A non-linear generalisation of the Loomis–Whitney inequality and applications" thought up?

Question

In Bennett, Carbery and Wright's paper A non-linear generalisation of the Loomis–Whitney inequality and applications, Claim 5 was used to generalise the case from characteristic functions to simple functions. I figured that through direct calculations, we can obtain

$$\int_U\displaystyle\prod_{j=1}^nf_j\circ\pi_j(x)dx\lesssim C\epsilon^{-\frac{1}{n-1}}N^n\displaystyle\prod_{j=1}^n\|f_j\|_{L^{n-1}(\mathbb{R}^{n-1})},$$

where $N$ is the maximum number of "steps" in each simple function, by the case of characteristic functions and Jensen's inequality.

In order to get rid of $N^n$, the authors used Claim 5 to raise the left hand side to arbitrary power $k\in\mathbb{N}$ and obtain

$$\left(\int_U\displaystyle\prod_{j=1}^nf_j\circ\pi_j(x)dx\right)^k\lesssim C^k\epsilon^{-\frac{k}{n-1}}(kN)^n\displaystyle\prod_{j=1}^n\|f_j\|_{L^{n-1}(\mathbb{R}^{n-1})}.$$

Letting $k\rightarrow\infty$, we can kill $N$.

My question is: how did they ever come up with this move? Is there a common technique or simply genius? To be honest, I would never think of raising to power $k$.

Answer 1

An instance of this idea of killing an unwanted factor in an inequality by considering an inequality for $k$-th powers and then taking the limit as $k\to\infty$ appears in the proof of the Kraft-McMillan inequality in information theory. The simpler case of prefix codes does not require the powers trick, but it is used in the proof for the more general case of uniquely decodable codes.

So yes, I would say it's a known technique and once you have seen a few examples of it, it may come more readily to mind as something to try in a new situation.

Answer 2

As Dan Romik points out, this technique is relatively old folklore by now. Terry Tao calls this the "tensor power trick" in a blog post dedicated to the subject; the two elementary applications he cites are $L^p$ interpolation and sumset bounds in additive combinatorics.

Statistical physicists have their own name for the technique (surprise, surprise): the "replica trick". There, most models of interest generalize to arbitrary dimensions straightforwardly, but observed quantities are typically analytic functions of modeled quantities. One copies the model in a new set of dimensions to compute arbitrary powers of the modeled quantities and then sums a Taylor series (or performs an equivalent limiting process).