  We consider Fourier multipliers in $\mathbb{R}^2$ of the form $m\circ\rho$
where $\rho$ is the Minkowski functional associated to a convex set in
$\mathbb{R}^2$, and prove $L^p$ bounds for the corresponding multiplier
operators. It is of interest to consider domains whose boundary is not smooth.
Our results depend on a notion of Minkowski dimension introduced by Seeger and
Ziesler that measures "flatness" of the boundary of the domain. Our methods
analyze the case of oscillatory multipliers
$\frac{e^{i\rho(\xi)}}{(1+|\xi|)^{-a}}$ associated to wave equations, which we
use to derive results for more general multiplier transformations.
