  Let ${\mathcal H}$ be a complex Hilbert space and let ${\mathcal B}({\mathcal
H})$ be the algebra of all bounded linear operators on ${\mathcal H}$. For a
positive integer $k$ less than the dimension of ${\mathcal H}$ and ${\mathbf A}
= (A_1, \dots, A_m)\in {\mathcal B}({\mathcal H})^m$, the joint $k$-numerical
range $W_k({\mathbf A})$ is the set of $(\alpha_1, \dots, \alpha_m) \in{\mathbb
C}^m$ such that $\alpha_i = \sum_{j = 1}^k \langle A_ix_j, x_j\rangle$ for an
orthonormal set $\{x_1, \ldots, x_k\}$ in ${\mathcal H}$. Relations between the
geometric properties of $W_k({\mathbf A})$ and the algebraic and analytic
properties of $A_1, \dots, A_m$ are studied. It is shown that there is $k\in
{\mathbb N}$ such that $W_k({\mathbf A})$ is a polyhedral set, i.e., the convex
hull of a finite set, if and only if $A_1, \dots, A_k$ have a common reducing
subspace ${\mathbf V}$ of finite dimension such that the compression of $A_1,
\dots, A_m$ on the subspace ${\mathbf V}$ are diagonal operators $D_1, \dots,
D_m$ and $W_k({\mathbf A}) = W_k(D_1, \dots, D_m)$. Characterization is also
given to ${\bf A}$ such that the closure of $W_k({\mathbf A})$ is polyhedral.
The conditions are related to the joint essential numerical range of ${\mathbf
A}$. These results are used to study ${\bf A}$ such that (a) $\{A_1, \dots,
A_m\}$ is a commuting family of normal operators, or (b) $W_k(A_1, \dots, A_m)$
is polyhedral for every positive integer $k$. It is shown that conditions (a)
and (b) are equivalent for finite rank operators but it is no longer true for
compact operators. Characterizations are given for compact operators $A_1,
\dots, A_m$ satisfying (a) and (b), respectively. Results are also obtained for
general non-compact operators.
