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

New $L^p$ bounds for Bochner-Riesz multipliers associated with convex planar domains with rough boundary