Let
H
n
{\mathcal {H}_n}
be the set of all
n
×
n
n \times n
hermitian matrices and
U
n
{\mathcal {U}_n}
the set of all
n
×
n
n \times n
unitary matrices. For any
c
=
(
c
1
,
…
,
c
n
)
∈
R
n
c = ({c_1}, \ldots ,{c_n}) \in {{\mathbf {R}}^n}
and
A
1
{A_1}
,
A
2
{A_2}
,
A
3
∈
H
n
{A_3} \in {\mathcal {H}_n}
, let
W
(
A
1
,
A
2
,
A
3
)
W({A_1},{A_2},{A_3})
denote the set
\[
{
(
tr
[
c
]
U
A
1
U
∗
,
tr
[
c
]
U
A
2
U
∗
,
tr
[
c
]
U
A
3
U
∗
)
:
U
∈
U
n
}
,
\{ ({\operatorname {tr}}[c]U{A_1}{U^*},{\operatorname {tr}}[c]U{A_2}{U^*},{\operatorname {tr}}[c]U{A_3}{U^*}):U \in {\mathcal {U}_n}\} ,
\]
where
[
c
]
[c]
is the diagonal matrix with
c
1
,
…
,
c
n
{c_1}, \ldots ,{c_n}
as diagonal entries. In this present note, the authors prove that if
n
>
2
n > 2
, then
W
c
(
A
1
,
A
2
,
A
3
)
{W_c}({A_1},{A_2},{A_3})
is always convex. Equivalent statements of this result, in terms of definiteness and inclusion relations, are also given. These results extend the theorems of Hausdorff-Toeplitz, Finsler and Westwick on numerical ranges, respectively.