Let $\Lambda$ be an $n$-dimensional lattice in $\mathbb R^n$ and let $\cal B$ be the set of all bases that generate $\Lambda$. For a basis $\mathbf{B}=[\mathbf{b}_1, ... ,\mathbf{b}_n]\in {\cal B}$, define $\mathbf{B}^\dagger = [\mathbf{b}^\dagger_1, ... ,\mathbf{b}^\dagger_n]$ to be the Gram-Schmidt orthogonalization of $\mathbf B$ (i.e., $\mathbf{B} = \mathbf{B}^\dagger\mathbf{U}$, for some upper-triangular matrix $\mathbf{U}$ with unit diagonal).

I'm interested in the *existence* of a lattice basis $\mathbf{B}\in{\cal B}$ that has the following property: $$||\mathbf{b}^\dagger_i||\leq||\mathbf{b}^\dagger_j||,\; \mbox{for all }\; 1\leq i < j \leq n.$$

Does such a basis exist for any $\Lambda$? Is this related to any problems already studied in the literature? Any pointers/insight will be very much appreciated. Thanks!