More Bisections by Hyperplane Arrangements

• Published in: Discrete & computational geometry Vol. 67; no. 1; pp. 33 - 64
• Main Authors: Blagojević, Pavle V. M., Blagojević, Aleksandra Dimitrijević, Karasev, Roman, Kliem, Jonathan
• Format: Journal Article
• Language: English
• Published: New York Springer US 2022; Springer Nature B.V
• Subjects: Collection; Combinatorics; Computational Mathematics and Numerical Analysis; Euclidean space; Funding; Geometry; Hyperplanes; Mathematics; Mathematics and Statistics; Polynomials
• ISSN: 0179-5376; 1432-0444
• DOI: 10.1007/s00454-021-00337-w

A union of an arrangement of affine hyperplanes
H
in
R
d
is the real algebraic variety associated to the principal ideal generated by the polynomial
p
H
given as the product of the degree one polynomials which define the hyperplanes of the arrangement. A finite Borel measure on
R
d
is bisected by the arrangement of affine hyperplanes
H
if the measure on the “non-negative side” of the arrangement
{
x
∈
R
d
:
p
H
(
x
)
≥
0
}
is the same as the measure on the “non-positive” side of the arrangement
{
x
∈
R
d
:
p
H
(
x
)
≤
0
}
. In 2017 Barba, Pilz & Schnider considered special, as well as modified cases of the following measure partition hypothesis: For a given collection of
j
finite Borel measures on
R
d
there exists a
k
-element affine hyperplane arrangement that bisects each of the measures into equal halves simultaneously. They showed that there are simultaneous bisections in the case when
d
=
k
=
2
and
j
=
4
. Furthermore, they conjectured that every collection of
j
measures on
R
d
can be simultaneously bisected with a
k
-element affine hyperplane arrangement provided that
d
≥
⌈
j
/
k
⌉
. The conjecture was confirmed in the case when
d
≥
j
/
k
=
2
a
by Hubard and Karasev in 2018. In this paper we give a different proof of the Hubard and Karasev result using the framework of Blagojević, Frick, Haase & Ziegler (2016), based on the equivariant relative obstruction theory of tom Dieck, which was developed for handling the Grünbaum–Hadwiger–Ramos hyperplane measure partition problem. Furthermore, this approach allowed us to prove even more, that for every collection of
2
a
(
2
h
+
1
)
+
ℓ
measures on 
R
2
a
+
ℓ
, where
1
≤
ℓ
≤
2
a
-
1
, there exists a
(
2
h
+
1
)
-element affine hyperplane arrangement that bisects all of them simultaneously. Our result was extended to the case of spherical arrangements and reproved by alternative methods in a beautiful way by Crabb [
8
].