  We begin a systematic study of the region of possible values of the volumes
of Minkowski subset sums of a collection of $M$ compact sets in $\mathbb{R}^d$,
which we call the Lyusternik region, and make some first steps towards
describing it. Our main result is that a fractional generalization of the
Brunn-Minkowski-Lyusternik inequality conjectured by Bobkov et al. (2011) holds
in dimension 1. Even though Fradelizi et al. (2016) showed that it fails in
general dimension, we show that a variant does hold in any dimension.
