  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 conjecture
due to Fuglede (1974) stated that $\Omega$ is a spectral set if and only if it
can tile the space by translations. While this conjecture was disproved for
general 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 fact
true.
  To the contrary, the "spectral implies tiling" direction of the conjecture
for 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 higher
dimensions, this direction of the conjecture remained completely open (even in
the case when $\Omega$ is a polytope) and could not be treated using the
previously developed techniques.
  In this paper we fully settle Fuglede's conjecture for convex bodies
affirmatively 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 polytope
which can tile the space by translations. To prove this we introduce a new
technique, involving a construction from crystallographic diffraction theory,
which allows us to establish a geometric "weak tiling" condition necessary for
a set $\Omega \subset \mathbb{R}^d$ to be spectral.

Classical Analysis and ODEs;Functional Analysis;Metric Geometry