A conjecture of Fuglede states that a bounded measurable
set $\Omega\subset\mathbb{R}^d$, of measure $1$, can tile
$\mathbb{R}^d$ by translations if and only if the Hilbert
space $L^2(\Omega)$ has an orthonormal basis consisting of
exponentials $e_\lambda(x) = \exp \{2\pi
i\langle{\lambda},{x}\rangle\}$. If $\Omega$ has the latter
property it is called {\em spectral}. Let $\Omega$ be a
polytope in $\mathbb{R}^d$ with the following property: there
is a direction $\xi \in S^{d-1}$ such that, of all the
polytope faces perpendicular to $\xi$, the total area of the
faces pointing in the positive $\xi$ direction is more than
the total area of the faces pointing in the negative $\xi$
direction. It is almost obvious that such a polytope $\Omega$
cannot tile space by translation. We prove in this paper that
such a domain is also not spectral, which agrees with
Fuglede's conjecture. As a corollary, we obtain a new proof of
the fact that a convex body that is spectral is necessarily
symmetric, in the case where the body is a polytope.