A Center Transversal Theorem for an Improved Rado Depth

• Published in: Discrete & computational geometry Vol. 60; no. 2; pp. 406 - 419
• Main Authors: Blagojević, Pavle V. M., Karasev, Roman, Magazinov, Alexander
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-09-2018; Springer Nature B.V
• Subjects: Collection; Combinatorics; Computational Mathematics and Numerical Analysis; Euclidean geometry; Euclidean space; Linear functions; Mathematics; Mathematics and Statistics; Center transversal; 52A35; Centerline; Rado theorem; 52C35; Tukey depth; 68U05
• ISSN: 0179-5376; 1432-0444
• DOI: 10.1007/s00454-018-0006-0

A celebrated result of Dol’nikov, and of Živaljević and Vrećica, asserts that for every collection of
m
measures
μ
1
,
⋯
,
μ
m
on the Euclidean space
R
n
+
m
-
1
there exists a projection onto an
n
-dimensional vector subspace
Γ
with a point in it at depth at least
1
/
(
n
+
1
)
with respect to each associated
n
-dimensional marginal measure
Γ
∗
μ
1
,
⋯
,
Γ
∗
μ
m
. In this paper we consider a natural extension of this result and ask for a minimal dimension of a Euclidean space in which one can require that for any collection of
m
measures there exists a vector subspace
Γ
with a point in it at depth slightly greater than
1
/
(
n
+
1
)
with respect to each
n
-dimensional marginal measure. In particular, we prove that if the required depth is
1
/
(
n
+
1
)
+
1
/
(
3
(
n
+
1
)
3
)
then the increase in the dimension of the ambient space is a linear function in both
m
and
n
.