This study will investigate the dimension of the kernel of a starshaped set, and the following result will be obtained: Let S be a compact set in some linear topological space L. For
1
⩽
k
⩽
n
1 \leqslant k \leqslant n
, the dimension of
ker
S
\ker S
is at least k if and only if for some
ε
>
0
\varepsilon > 0
and some n-dimensional flat
F
n
{F^n}
in L, every
f
(
n
,
k
)
f(n,k)
points of S see via S a common k-dimensional neighborhood in
F
n
{F^n}
having radius
ε
\varepsilon
. The number
f
(
n
,
k
)
f(n,k)
is defined inductively as follows:
\[
f
(
2
,
1
)
=
4
,
f
(
n
,
k
)
=
f
(
n
−
1
,
k
)
+
n
+
2
for
3
⩽
n
and
1
⩽
k
⩽
n
−
1
,
f
(
n
,
n
)
=
n
+
1.
\begin {array}{*{20}{c}} {f(2,1) = 4,} \hfill \\ {f(n,k) = f(n - 1,k) + n + 2\quad {\text {for}}\;3 \leqslant n\;{\text {and}}\;1 \leqslant k \leqslant n - 1,} \hfill \\ {f(n,n) = n + 1.} \hfill \\ \end {array}
\]