Enclosing Depth and Other Depth Measures
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Enclosing Depth and Other Depth Measures. / Schnider, Patrick.
In: Combinatorica, Vol. 43, No. 5, 2023, p. 1007-1029.Research output: Contribution to journal › Journal article › Research › peer-review
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TY - JOUR
T1 - Enclosing Depth and Other Depth Measures
AU - Schnider, Patrick
N1 - Publisher Copyright: © 2023, The Author(s).
PY - 2023
Y1 - 2023
N2 - We study families of depth measures defined by natural sets of axioms. We show that any such depth measure is a constant factor approximation of Tukey depth. We further investigate the dimensions of depth regions, showing that the Cascade conjecture, introduced by Kalai for Tverberg depth, holds for all depth measures which satisfy our most restrictive set of axioms, which includes Tukey depth. Along the way, we introduce and study a new depth measure called enclosing depth, which we believe to be of independent interest, and show its relation to a constant-fraction Radon theorem on certain two-colored point sets.
AB - We study families of depth measures defined by natural sets of axioms. We show that any such depth measure is a constant factor approximation of Tukey depth. We further investigate the dimensions of depth regions, showing that the Cascade conjecture, introduced by Kalai for Tverberg depth, holds for all depth measures which satisfy our most restrictive set of axioms, which includes Tukey depth. Along the way, we introduce and study a new depth measure called enclosing depth, which we believe to be of independent interest, and show its relation to a constant-fraction Radon theorem on certain two-colored point sets.
KW - Combinatorial depth measures
KW - Discrete geometry
KW - Topological methods
U2 - 10.1007/s00493-023-00045-4
DO - 10.1007/s00493-023-00045-4
M3 - Journal article
AN - SCOPUS:85163127457
VL - 43
SP - 1007
EP - 1029
JO - Combinatorica
JF - Combinatorica
SN - 0209-9683
IS - 5
ER -
ID: 359597345