The challenge of taking many variables into account in optimization problems\nmay be overcome under the hypothesis of low effective dimensionality. Then, the\nsearch of solutions can be reduced to the random embedding of a low dimensional\nspace into the original one, resulting in a more manageable optimization\nproblem. Specifically, in the case of time consuming black-box functions and\nwhen the budget of evaluations is severely limited, global optimization with\nrandom embeddings appears as a sound alternative to random search. Yet, in the\ncase of box constraints on the native variables, defining suitable bounds on a\nlow dimensional domain appears to be complex. Indeed, a small search domain\ndoes not guarantee to find a solution even under restrictive hypotheses about\nthe function, while a larger one may slow down convergence dramatically. Here\nwe tackle the issue of low-dimensional domain selection based on a detailed\nstudy of the properties of the random embedding, giving insight on the\naforementioned difficulties. In particular, we describe a minimal\nlow-dimensional set in correspondence with the embedded search space. We\nadditionally show that an alternative equivalent embedding procedure yields\nsimultaneously a simpler definition of the low-dimensional minimal set and\nbetter properties in practice. Finally, the performance and robustness gains of\nthe proposed enhancements for Bayesian optimization are illustrated on\nnumerical examples.\n