Let
x
1
x_1
,
x
2
x_2
, …,
x
n
x_n
be
n
n
points on the sphere
S
2
S^2
. Determining the value
inf
∑
1
≤
k
>
j
≤
n
|
x
k
−
x
j
|
−
1
\inf \sum _{1\leq k>j\leq n}|x_k-x_j|^{-1}
, is a long-standing open problem in discrete geometry, which is known as Thomson’s problem. In this paper, we propose a reverse problem on the sphere
S
d
−
1
S^{d-1}
in
d
d
-dimensional Euclidean space, which is equivalent to establish the reverse Thomson inequality. In the planar case, we establish two variants of the reverse Thomson inequality.
In addition, we give a proof to the minimal logarithmic energy of
x
1
x_1
,
x
2
x_2
, …,
x
n
x_n
and two dimensional Thomson’s problem on the unit circle for all integer
n
≥
2
n\geq 2
.