## Multilinear geometric inequalities

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27/11/2021##### Item status

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27/11/2022##### Author

McIntyre, Finlay

Dupree McIntyre, Finlay

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##### Abstract

In this thesis we explore a multilinear duality principle of Carbery–Hänninen–Valdimarsson
for multilinear geometric inequalities arising in Harmonic Analysis. In Chapter 2, we give a
brief survey of how this theory manifests for a number of classical inequalities. The material in
Chapter 2 is mostly expository except for two original results: Theorem 2.17, which gives an
explicit factorisation of the classical Young convolution inequality on R, and Proposition 2.10,
establishing the equivalence of two natural formulations of an affine-invariant Finner inequality.
The material in Section 2.3 is also new.
In Chapter 3, we discuss a certain structural duality between instances of the geometric
Brascamp–Lieb inequality, and briefly explore how this manifests on the level of factorisations.
Most of the material presented in this chapter is not original research. However, the connection between Lemma 3.2 and Proposition 3.7 in the setting of the geometric Brascamp–Lieb
inequality is new. In particular, we give a new proof for a special case of Proposition 3.7, which
only relies on the sharpness of the geometric Brascamp–Lieb inequality along with a generalised
version of Lemma 3.2.
In Chapter 4 we study factorisations due to Guth for the endpoint multilinear Kakeya
inequality, and factorisations due to Zhang for a corresponding generalisation to kj -planes. In
particular, we uncover interesting connections to classical notions of convex geometry. The
main contributions of this chapter stem from the statements and proofs of Theorems 4.7 and
4.13. In particular, these two geometric observations allow us to translate Guth’s visibility
theorem, stated below as Theorem 4.33, into statements about convex bodies, which lead us to
the formulation of Theorems 4.8 and 4.14.
In Chapter 5 we study a sequence of generalised versions of Guth’s directed volume - see
Definition 4.3. The main contribution of this chapter is Theorem 5.1, giving a sharp chain of
inequalities for such geometric quantities. Furthermore, we conjecture restricted inequalities of
the same type, which are optimal for boundaries of parallelotopes and related to a vector-valued
generalisation of the classical Maclaurin inequality. In Chapter 6, we prove various special cases
of this vector-valued variant of the Maclaurin inequality, and highlight connections to open
problems in convex geometry.