Let
S
S
be a compact connected set in
R
2
{R^2}
,
S
S
not convex. Then
S
S
is starshaped if and only if every 3 points of local nonconvexity of
S
S
are clearly visible from a common point of
S
S
. For
k
=
1
k = 1
or
k
=
2
k = 2
, dimker
S
⩾
S \geqslant
k
k
if and only if for some
∈>
0
\in > 0
, every
f
(
k
)
=
max
{
3
,
6
−
2
k
}
f(k) = \max \left \{ {3,6 - 2k} \right \}
points of local nonconvexity of
S
S
are clearly visible from a common
k
k
-dimensional
∈
\in
neighborhood in
S
S
. Each result is best possible.