Let
\Omega_k
Ω
k
denote the collection of all nonempty closed convex subsets of
\mathbb{R}^k
R
k
. We provide short proofs for the following: (i)
\{x\in {\mathbb{R}}^k:dist(x,A)=\varepsilon\}
{
x
∈
R
k
:
d
i
s
t
(
x
,
A
)
=
ε
}
is a
C^1
C
1
-manifold of dimension
k-1
k
−
1
for every
A\in \Omega_k\setminus \{\mathbb{R}^k\}
A
∈
Ω
k
∖
{
R
k
}
and
\varepsilon>0
ε
>
0
, (ii)
\{x\in {\mathbb{R}}^k:dist(x,A)=dist(x,B)\}
{
x
∈
R
k
:
d
i
s
t
(
x
,
A
)
=
d
i
s
t
(
x
,
B
)
}
is a
C^1
C
1
-manifold of dimension
k-1
k
−
1
for any two disjoint
A, B\in \Omega_k
A
,
B
∈
Ω
k
. We also study the distance of points in
\mathbb{R}^k
R
k
to finitely many closed convex sets. Let
k,n\ge 2
k
,
n
≥
2
and
A=\bigcup_{j=1}^n A_j
A
=
⋃
j
=
1
n
A
j
, where
A_1,\ldots,A_n\in \Omega_k
A
1
,
…
,
A
n
∈
Ω
k
are pairwise disjoint. We consider a Voronoi type decomposition of
\mathbb{R}^k
R
k
and establish some topological properties of its ‘conflict set’. Letting
X_p=\{x\in {\mathbb{R}}^k:|\{a\in A: \|x-a\| =dist(x,A)\}|=p\}
X
p
=
{
x
∈
R
k
:
∣
{
a
∈
A
:
∥
x
−
a
∥
=
d
i
s
t
(
x
,
A
)
}
∣
=
p
}
, we prove with the help of result (ii) stated above that
X_1\cup X_2
X
1
∪
X
2
is a connected dense open subset of
\mathbb{R}^k
R
k
and that
\overline{X_2}=\bigcup_{p=2}^n X_p
X
2
‾
=
⋃
p
=
2
n
X
p
.