In 1960 Grünbaum asked whether for any finite mass in
R
d
\mathbb {R}^d
there are
d
d
hyperplanes that cut it into
2
d
2^d
equal parts. This was proved by Hadwiger (1966) for
d
≤
3
d\le 3
, but disproved by Avis (1984) for
d
≥
5
d\ge 5
, while the case
d
=
4
d=4
remained open.
More generally, Ramos (1996) asked for the smallest dimension
Δ
(
j
,
k
)
\Delta (j,k)
in which for any
j
j
masses there are
k
k
affine hyperplanes that simultaneously cut each of the masses into
2
k
2^k
equal parts. At present the best lower bounds on
Δ
(
j
,
k
)
\Delta (j,k)
are provided by Avis (1984) and Ramos (1996), the best upper bounds by Mani-Levitska, Vrećica and Živaljević (2006). The problem has been an active testing ground for advanced machinery from equivariant topology.
We give a critical review of the work on the Grünbaum–Hadwiger–Ramos problem, which includes the documentation of essential gaps in the proofs for some previous claims. Furthermore, we establish that
Δ
(
j
,
2
)
=
1
2
(
3
j
+
1
)
\Delta (j,2)= \frac 12(3j+1)
in the cases when
j
−
1
j-1
is a power of
2
2
,
j
≥
5
j\ge 5
.