Let
K
K
and
K
0
K_0
be convex bodies in
R
d
\mathbb {R}^d
, such that
K
K
contains the origin, and define the process
(
K
n
,
p
n
)
(K_n, p_n)
,
n
≥
0
n \geq 0
, as follows: let
p
n
+
1
p_{n+1}
be a uniform random point in
K
n
K_n
, and set
K
n
+
1
=
K
n
∩
(
p
n
+
1
+
K
)
K_{n+1} = K_n \cap (p_{n+1} + K)
. Clearly,
(
K
n
)
(K_n)
is a nested sequence of convex bodies which converge to a non-empty limit object, again a convex body in
R
d
\mathbb {R}^d
. We study this process for
K
K
being a regular simplex, a cube, or a regular convex polygon with an odd number of vertices. We also derive some new results in one dimension for non-uniform distributions.