doi:10.1112/jlms.12411engAlonso Gutiérrez, DavidBernués, JulioBrazitikos, SilouanosCarbery, AnthonyOn affine invariant and local Loomis–Whitney type inequalitiesART-2020-120845We prove various extensions of the Loomis–Whitney inequality and its dual, where the subspaces on which the projections (or sections) are considered are either spanned by vectors $w_i$ of a not necessarily orthonormal basis of $\mathbb{R^n}$ , or their orthogonal complements. In order to prove such inequalities, we estimate the constant in the Brascamp–Lieb inequality in terms of the vectors $w_i$ . Restricted and functional versions of the inequality will also be considered.2020http://zaguan.unizar.es/record/9703410.1112/jlms.12411http://zaguan.unizar.es/record/97034oai:zaguan.unizar.es:97034info:eu-repo/grantAgreement/ES/DGA/E64info:eu-repo/grantAgreement/ES/MINECO/MTM2016-77710-PJOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES 103, 4 (2020), 1377-1401byhttp://creativecommons.org/licenses/by/3.0/es/info:eu-repo/semantics/openAccess