  From a geometric point of view, Pauli's exclusion principle defines a
hypersimplex. This convex polytope describes the compatibility of $1$-fermion
and $N$-fermion density matrices, therefore it coincides with the convex hull
of the pure $N$-representable $1$-fermion density matrices. Consequently, the
description of ground state physics through $1$-fermion density matrices may
not necessitate the intricate pure state generalized Pauli constraints. In this
article, we study the generalization of the $1$-body $N$-representability
problem to ensemble states with fixed spectrum $\mathbf{w}$, in order to
describe finite-temperature states and distinctive mixtures of excited states.
By employing ideas from convex analysis and combinatorics, we present a
comprehensive solution to the corresponding convex relaxation, thus
circumventing the complexity of generalized Pauli constraints. In particular,
we adapt and further develop tools such as symmetric polytopes, sweep
polytopes, and Gale order. For both fermions and bosons, generalized exclusion
principles are discovered, which we determine for any number of particles and
dimension of the $1$-particle Hilbert space. These exclusion principles are
expressed as linear inequalities satisfying hierarchies determined by the
non-zero entries of $\mathbf{w}$. The two families of polytopes resulting from
these inequalities are part of the new class of so-called lineup polytopes.
