  Mixed volumes in $n$-dimensional Euclidean space are functionals of
$n$-tuples consisting of convex bodies $K,L,C_1,\ldots,C_{n-2}$. The
Alexandrov--Fenchel inequalities are fundamental inequalities between mixed
volumes of convex bodies, which cover as very special cases many important
inequalities between basic geometric functionals. The problem of characterizing
completely the equality cases in the Alexandrov--Fenchel inequality is wide
open. Major recent progress was made by Yair Shenfeld and Ramon van Handel
\cite{SvH22,SvH23+}, in particular they resolved the problem in the cases where
$K,L$ are general convex bodies and $C_1,\ldots,C_{n-2}$ are polytopes, zonoids
or smooth bodies (under some dimensional restriction). We introduce the class
of polyoids, which includes polytopes, zonoids and triangle bodies, and
characterize polyoids by using generating measures. Based on this
characterization and Shenfeld and van Handel's contribution, we extend their
result to a class of convex bodies containing all polyoids and smooth bodies.
Our result is stated in terms of the support of the mixed area measure of the
unit ball $B^n$ and $C_1,\ldots,C_{n-2}$. A geometric description of this
support is provided in the accompanying work \cite{HugReichert23+}.
