  We extend the estimates for maximal Fourier restriction operators proved by
M\"{u}ller, Ricci, and Wright in \cite{MR3960255} and Ramos in \cite{MR4055940}
to the case of arbitrary convex curves in the plane, with constants uniform in
the curve. The improvement over M\"{u}ller, Ricci, and Wright and Ramos is
given by the removal of the $\mathcal{C}^2$ regularity condition on the curve.
This requires the choice of an appropriate measure for each curve, that is
suggested by an affine invariant construction of Oberlin in \cite{MR1960918}.
As corollaries, we obtain a uniform Fourier restriction theorem for arbitrary
convex curves, and a result on the Lebesgue points of the Fourier transform on
the curve.
