## Abstract

Given a graph H on vertex set {1,2,⋯,n} and a function f:[0,1]^{2}→R, define ‖f‖_{H}:=|∫∏ij∈E(H)f(x_{i},x_{j})dμ^{|V(H)|}|^{1/|E(H)|} 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.

Original language | English |
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Pages (from-to) | 130-165 |

Number of pages | 36 |

Journal | Advances in Mathematics |

Volume | 315 |

DOIs | |

Publication status | Published - 2017 Jul 31 |

### Bibliographical note

Publisher Copyright:© 2017 Elsevier Inc.

## All Science Journal Classification (ASJC) codes

- General Mathematics