  Mixed volumes in $n$-dimensional Euclidean space are functionals of
$n$-tuples of convex bodies $K,L,C_1,\ldots,C_{n-2}$. The Alexandrov--Fenchel
inequalities are fundamental inequalities between mixed volumes of convex
bodies. As very special cases they cover or imply many important inequalities
between basic geometric functionals. A complete characterization of the
equality cases in the Alexandrov--Fenchel inequality remains a challenging open
problem. 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
$C_1,\ldots,C_{n-2}$ are polytopes, zonoids or smooth bodies (under some
dimensional restriction). In \cite{HugReichert23+} we introduced the class of
polyoids, which are defined as limits of finite Minkowski sums of polytopes
having a bounded number vertices. Polyoids encompass polytopes, zonoids and
triangle bodies, and they can be characterized by means of generating measures.
Based on this characterization and Shenfeld and van Handel's contribution, we
extended their result to polyoids (or smooth bodies). Our previous result was
stated in terms of the support of the mixed area measure associated with the
unit ball $B^n$ and $C_1,\ldots,C_{n-2}$. This characterization result is
completed in the present work which more generally provides a geometric
description of the support of the mixed area measure of an arbitrary
$(n-1)$-tuple of polyoids (or smooth bodies). The result confirms a
long-standing conjecture by Rolf Schneider in the case of polyoids and hence,
in particular, of zonoids.
