Aarhus University Seal / Aarhus Universitets segl

Uniqueness of the measurement function in Crofton's formula

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaperJournal articleResearchpeer-review

Standard

Uniqueness of the measurement function in Crofton's formula. / Eriksen, Rikke; Kiderlen, Markus.

In: Advances in Applied Mathematics, Vol. 116, 102004, 05.2020.

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaperJournal articleResearchpeer-review

Harvard

APA

CBE

MLA

Vancouver

Author

Bibtex

@article{60e9fdc030074d888e4b8be3d4bce4f8,
title = "Uniqueness of the measurement function in Crofton's formula",
abstract = "Crofton's intersection formula states that the (n−j)th intrinsic volume of a compact convex set in Rn can be obtained as an invariant integral of the (k−j)th intrinsic volume of sections with k-planes. This paper discusses the question if the (k−j)th intrinsic volume can be replaced by other functionals, that is, if the measurement function in Crofton's formula is unique. The answer is negative: we show that the sums of the (k−j)th intrinsic volume and certain translation invariant continuous valuations of homogeneity degree k yield counterexamples. If the measurement function is local, these turn out to be the only examples when k=1 or when k=2 and we restrict considerations to even measurement functions. Additional examples of local functionals can be constructed when k≥2.",
keywords = "Crofton's formula, Klain functional, Local functions, Spherical lifting, Uniqueness, Valuation",
author = "Rikke Eriksen and Markus Kiderlen",
year = "2020",
month = may,
doi = "10.1016/j.aam.2020.102004",
language = "English",
volume = "116",
journal = "Advances in Applied Mathematics",
issn = "0196-8858",
publisher = "Academic Press",

}

RIS

TY - JOUR

T1 - Uniqueness of the measurement function in Crofton's formula

AU - Eriksen, Rikke

AU - Kiderlen, Markus

PY - 2020/5

Y1 - 2020/5

N2 - Crofton's intersection formula states that the (n−j)th intrinsic volume of a compact convex set in Rn can be obtained as an invariant integral of the (k−j)th intrinsic volume of sections with k-planes. This paper discusses the question if the (k−j)th intrinsic volume can be replaced by other functionals, that is, if the measurement function in Crofton's formula is unique. The answer is negative: we show that the sums of the (k−j)th intrinsic volume and certain translation invariant continuous valuations of homogeneity degree k yield counterexamples. If the measurement function is local, these turn out to be the only examples when k=1 or when k=2 and we restrict considerations to even measurement functions. Additional examples of local functionals can be constructed when k≥2.

AB - Crofton's intersection formula states that the (n−j)th intrinsic volume of a compact convex set in Rn can be obtained as an invariant integral of the (k−j)th intrinsic volume of sections with k-planes. This paper discusses the question if the (k−j)th intrinsic volume can be replaced by other functionals, that is, if the measurement function in Crofton's formula is unique. The answer is negative: we show that the sums of the (k−j)th intrinsic volume and certain translation invariant continuous valuations of homogeneity degree k yield counterexamples. If the measurement function is local, these turn out to be the only examples when k=1 or when k=2 and we restrict considerations to even measurement functions. Additional examples of local functionals can be constructed when k≥2.

KW - Crofton's formula

KW - Klain functional

KW - Local functions

KW - Spherical lifting

KW - Uniqueness

KW - Valuation

UR - http://www.scopus.com/inward/record.url?scp=85078669131&partnerID=8YFLogxK

U2 - 10.1016/j.aam.2020.102004

DO - 10.1016/j.aam.2020.102004

M3 - Journal article

AN - SCOPUS:85078669131

VL - 116

JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

SN - 0196-8858

M1 - 102004

ER -