  In this paper we introduce bilinear Bochner-Riesz means associated with
convex domains in the plane $\mathbb R^2$ and study their $L^p-$boundedness
properties for a wide range of exponents. One of the important aspects of our
proof involves the use of bilinear Kakeya maximal function in the context of
bilinear Bochner-Riesz problem. This amounts to establish suitable $L^p-$
estimates for the later. We also point out some natural connections between
bilinear Kakeya maximal function and Lacey's bilinear maximal function.
