Given a graph H on vertex set {1,2,⋯,n} and a function f:[0,1]2→R, define \begin{align*} \|f\|_{H}:=\left\vert\int
\prod_{ij\in E(H)}f(x_i,x_j)d\mu^{|V(H)|}\right\vert^{1/|E(H)|}, \end{align*}
where μ is the Lebesgue measure on [0,1]. We say that H is norming if
∥⋅∥H is a semi-norm. A similar notion ∥⋅∥r(H) is defined by
∥f∥r(H):=∥∣f∣∥H and H is said to be weakly norming if
∥⋅∥r(H) is a norm. Classical results show that weakly norming graphs
are necessarily bipartite. In the other direction, Hatami showed that even
cycles, complete bipartite graphs, and hypercubes are all weakly norming. We
demonstrate that any graph whose edges percolate in an appropriate way under
the action of a certain natural family of automorphisms is weakly norming. This
result includes all previously known examples of weakly norming graphs, but
also allows us to identify a much broader class arising from finite reflection
groups. We include several applications of our results. In particular, we
define and compare a number of generalisations of Gowers' octahedral norms and
we prove some new instances of Sidorenko's conjecture.Comment: 29 page