In this paper we study the so-called Radon inversion problem in bounded, circular, strictly convex domains with $${\mathcal {C}}^2$$
C
2
boundary. We show that given $$p>0$$
p
>
0
and a strictly positive, continuous function $$\Phi $$
Φ
on $$\partial \Omega $$
∂
Ω
, by use of homogeneous polynomials it is possible to construct a holomorphic function $$f \in {\mathcal {O}}(\Omega )$$
f
∈
O
(
Ω
)
such that $$\displaystyle \smallint _0^1 |f(zt)|^pdt = \Phi (z)$$
∫
0
1
|
f
(
z
t
)
|
p
d
t
=
Φ
(
z
)
for all $$z \in \partial \Omega $$
z
∈
∂
Ω
. In our approach we make use of so-called lacunary K-summing polynomials (see definition below) that allow us to construct solutions with in some sense extremal properties.