The Brunn-Minkowski Theory has seen several generalizations over the past century. Many of the core ideas have been generalized to measures. With the goal of framing these generalizations as a weighted Brunn-Minkowski theory, we prove the Minkowski existence theorem for a large class of Borel measures with continuous density, denoted by Λn: for ν a finite, even Borel measure on the unit sphere and even μ∈Λn, there exists a symmetric convex body
Nearby in the stack
K
such that
dν(u)=cμ,KdSKμ(u),
where
cμ,K
is a quantity that depends on
μ
and
K
and
dSKμ(u)
is the surface area-measure of
K
with respect to
μ
. Examples of measures in
Λn
are homogeneous measures (with
cμ,K=1
) and probability measures with radially decreasing densities (e.g. the Gaussian measure). We will also consider weighted projection bodies
ΠμK
by classifying them and studying the isomorphic Shephard problem: if
μ
and
ν
are even, homogeneous measures with density and
K
and
L
are symmetric convex bodies such that
ΠμK⊂ΠνL
, then can one find an optimal quantity
A>0
such that
μ(K)≤Aν(L)
? Among other things, we show that, in the case where