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A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space$L^2(\Omega)$ has an orthogonal basis of exponential functions. A conjecturedue to Fuglede (1974) stated that $\Omega$ is a spectral set if and only if itcan tile the space by translations. While this conjecture was disproved forgeneral sets, it has long been known that for a convex body $\Omega \subset\mathbb{R}^d$ the "tiling implies spectral" part of the conjecture is in facttrue.  To the contrary, the "spectral implies tiling" direction of the conjecturefor convex bodies was proved only in $\mathbb{R}^2$, and also in $\mathbb{R}^3$under the a priori assumption that $\Omega$ is a convex polytope. In higherdimensions, this direction of the conjecture remained completely open (even inthe case when $\Omega$ is a polytope) and could not be treated using thepreviously developed techniques.  In this paper we fully settle Fuglede's conjecture for convex bodiesaffirmatively in all dimensions, i.e. we prove that if a convex body $\Omega\subset \mathbb{R}^d$ is a spectral set then $\Omega$ is a convex polytopewhich can tile the space by translations. To prove this we introduce a newtechnique, involving a construction from crystallographic diffraction theory,which allows us to establish a geometric "weak tiling" condition necessary fora set $\Omega \subset \mathbb{R}^d$ to be spectral.

The Fuglede conjecture for convex domains is true in all dimensions