  We consider generalized Bochner-Riesz multipliers of the form
$(1-\rho(\xi))_+^{\lambda}$ where $\rho(\xi)$ is the Minkowski functional of a
convex domain in $\mathbb{R}^2$, with emphasis on domains for which the usual
Carleson-Sj\"{o}lin $L^p$ bounds can be improved. We produce convex domains for
which previous results due to Seeger and Ziesler are not sharp. We identify two
key properties of convex domains that lead to improved $L^p$ bounds for the
associated Bochner-Riesz operators. First, we introduce the notion of the
"additive energy" of the boundary of a convex domain. Second, we associate a
set of directions to a convex domain and define a sequence of Nikodym-type
maximal operators corresponding to this set of directions. We show that domains
that have low higher order additive energy, as well as those which have
asymptotically good $L^q$ bounds for the corresponding sequence of Nikodym-type
maximal operators where $q=(p^{\prime}/2)^{\prime}$, have improved $L^p$ bounds
for the associated Bochner-Riesz operators over those proved by Seeger and
Ziesler.
