We introduce a class of convex equivolume partitions, among this class, there exists an optimal partition manner. Expected $L_2-$discrepancy are discussed under these partitions. There are two main results. First, under this kind of partitions, we generate random point sets with smaller expected $L_2-$discrepancy than classical jittered sampling for the same sampling number. Further, among these new partitions, there is the optimal partition for the expected $L_2-$discrepancy. Second, an explicit expected $L_2-$discrepancy upper bound under this kind of partitions is also given.

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