  We propose to derive deviation measures through the Minkowski gauge of a
given set of acceptable positions. We show that, given a suitable acceptance
set, any positive homogeneous deviation measure can be accommodated in our
framework. In doing so, we provide a new interpretation for such measures,
namely, that they quantify how much one must shrink or deleverage a position
for it to become acceptable. In particular, the Minkowski Deviation of a set
which is convex, stable under scalar addition, and radially bounded at
non-constants, is a generalized deviation measure. Furthermore, we explore the
relations existing between mathematical and financial properties attributable
to an acceptance set, and the corresponding properties of the induced measure.
Hence, we fill the gap that is the lack of an acceptance set for deviation
measures. Dual characterizations in terms of polar sets and support functionals
are provided.
