  In this article, we investigate the boundedness of the bilinear Bochner-Riesz
means $S^{\alpha }$ in the non-Banach triangle case. We improve the
corresponding results in [Bern] in two aspects: Our partition of the non-Banach
triangle is simpler and we obtain lower smoothness indices $\alpha
(p_{1},p_{2})$ for various cases apart from $1 \leq p_1=p_2 <2$.
