It is well known that a nondegenerate projective subvariety $${X \subset \mathbb {P}^r}$$ of degree d and codimension c > 1 has minimal degree (i.e., d = c + 1) if and only if index( X) ≥ c if and only if X has no multisecant c-space. In this paper we extend this result by classifying varieties with index( X) ≥ c − s or with no multisecant ( c − s)-space for s = 1 and 2.