Abstract. Let V = V (n, q) denote the vector space of dimension n over the finite field with q elements. A subspace partition P of V is a collection of nontrivial subspaces of V such that each nonzero vector of V is in exactly one subspace of P. For any integer d, the d-supertail of P is the set of subspaces in P of dimension less than d, and it is denoted by ST . Let σ q (n, t) denote the minimum number of subspaces in any subspace partition of V in which the largest subspace has dimension t. It was shown by Heden et al. that |ST | ≥ σ q (d, t), where t is the largest dimension of a subspace in ST . In this paper, we show that if |ST | = σ q (d, t), then the union of all the subspaces in ST constitutes a subspace under certain conditions.