For weighted Riesz potentials of the form
K
(
x
,
y
)
=
w
(
x
,
y
)
/
K(x,y)=w(x,y)/
|
x
−
y
|
s
|x-y|^s
, we investigate
N
N
-point configurations
x
1
,
x
2
,
…
,
x
N
x_1,x_2, \ldots , x_N
on a
d
d
-dimensional compact subset
A
A
of
R
p
\mathbb {R}^p
for which the minimum of
∑
j
=
1
N
K
(
x
,
x
j
)
\sum _{j=1}^NK(x,x_j)
on
A
A
is maximal. Such quantities are called
N
N
-point Riesz
s
s
-polarization (or Chebyshev) constants. For
s
⩾
d
s\geqslant d
, we obtain the dominant term as
N
→
∞
N\to \infty
of such constants for a class of
d
d
-rectifiable subsets of
R
p
\mathbb {R}^p
. This class includes compact subsets of
d
d
-dimensional
C
1
C^1
manifolds whose boundary relative to the manifold has
d
d
-dimensional Hausdorff measure zero, as well as finite unions of such sets when their pairwise intersections have measure zero. We also explicitly determine the weak-star limit distribution of asymptotically optimal
N
N
-point configurations for weighted
s
s
-polarization as
N
→
∞
N\to \infty
.