This paper focuses on borders and lossless representations for Constrained Datacubes of database relations, which can represent many-valued contexts. The final goal is to optimize both storage space and computation time. First we study the succinct representation through the borders Lower / Upper and Upper (^sharp) / Upper. However, these borders are not information-lossless. Therefore, by using the concept of cube closure, from a FCA perspective, we define three new information lossless representations for Constrained Datacubes: the L-Constrained Closed Datacube, the (U^sharp)-Constrained Closed Datacube and the (U^{sharpsharp})-Constrained Closed Cube. The latter is obtained by using the constrained closed tuples along with the border Upper (^sharp) from which redundancy is discarded in order to obtain an optimal representation. Finally, we evaluate experimentally the size of the introduced representations. The results are strongly emphasizing the idea that the latest structure is, to the best of our knowledge, the smallest representation of Constrained Datacubes.