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2016

An improvement on the Rado bound for the centerline depth

Magazinov, Alexander, Pór, Attila

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Understand

Let $\mu$ be a Borel probability measure in $\mathbb R^d$.

For a $k$-flat $\alpha$ consider the value $\inf \mu(H)$, where $H$ runs through all half-spaces containing $\alpha$.
This infimum is called the half-space depth of $\alpha$.
Bukh, Matou\v{s}ek and Nivasch conjectured that for every $\mu$ and every $0 \leq k < d$ there exists a $k$-flat with the depth at least $\tfrac{k + 1}{k + d + 1}$.

An improvement on the Rado bound for the centerline depth · Around

Built on

R. Rado, A theorem on general measure,
1946
Earlier work this paper cites.
J.W. Milnor, and J.D. Stasheff, Characteristic Classes,
1974
Earlier work this paper cites.
E. Boros, and Z. Füredi, The number of triangles covering the center of an n-set,
1984
Earlier work this paper cites.
R.J. Wilson, Introduction to Graph Theory, 4th Edition,
1996
Earlier work this paper cites.

Similar

A. Hatcher, Algebraic Topology,
2001
Cited alongside, same era.
B. Bukh, J. Matoušek, and G. Nivasch, Stabbing simplices by points and flats,
2010
Cited alongside, same era.
B. Klartag, On nearly radial marginals of high-dimensional probability measures,
2010
Cited alongside, same era.

Then

H.L. Royden, and P.M. Fitzpatrick, Real Analysis, 4th Edition,
2010
Later among the works it cites.
B. Bukh, and G. Nivasch, Upper bounds for centerlines,
2012
Later among the works it cites.
D. Eisenbud, and J. Harris, 3264 and All That. A Second Course in Algebraic Geometry,
2016
Closest in time.

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