# Reverse Loomis-Whitney Inequality for functions

I was wondering if the reverse Loomis-Whitney inequality holds for general functions:

Let $$n\geq 2$$. Let $$(X_i,\mu_i)$$, $$1\leq i\leq n$$ be measure spaces. Write $$x=(x_1,\dots,x_n)$$ and for each $$1\leq i\leq n$$, write $$\pi_i(x)=x_i'=(x_1,\dots,x_{i-1},x_{i+1},\dots,x_n)\in X'_i:=X_1\times \cdots X_{i-1}\times X_{i+1}\times\cdots X_n.$$ Let $$f_i$$ be nonnegative measurable functions defined on $$X'_i$$. Then do we have the following inequality: $$\gamma_n\prod_{i=1}^n \left\| f_i\right\|_{L^{n-1}(X'_i)}\leq \int_{X_1}\cdots \int_{X_n}\prod_{i=1}^n f_i(x_i')dx.$$ where $$0<\gamma_n<1$$ is an absolute constant?

It is known that if $$f_i=1_{\pi_i(K)}$$ for some compact set $$K$$, then the reverse inequality holds. (See, for example, https://arxiv.org/pdf/1607.07891.pdf)

This conjecture is trivially true for $$n=2$$, but false for any $$n\ge3$$. Indeed, take any $$n\ge3$$ and for all $$i$$ let $$X_i=\mathbb R$$ and $$\mu_i=\lambda$$, the Lebesgue measure. Next, let $$f_i(x'_i):=\prod_{j:\ j\ne i}1_{i Then $$\prod_i f_i(x'_i)=\prod_i\prod_{j:\ j\ne i}1_{i for all $$x$$, but $$\|f_i\|_1=\int f_i=\prod_{j:\ j\ne i}\int_i^{i+1}dx_j=1$$ for all $$i$$. So, the right side of your inequality is $$0$$, whereas its left side is $$\gamma_n>0$$.