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Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaper › Journal article › Research › peer-review

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

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaper › Journal article › Research › peer-review

Eriksen, R & Kiderlen, M 2020, 'Uniqueness of the measurement function in Crofton's formula', *Advances in Applied Mathematics*, vol. 116, 102004. https://doi.org/10.1016/j.aam.2020.102004

Eriksen, R., & Kiderlen, M. (2020). Uniqueness of the measurement function in Crofton's formula. *Advances in Applied Mathematics*, *116*, [102004]. https://doi.org/10.1016/j.aam.2020.102004

Eriksen R, Kiderlen M. 2020. Uniqueness of the measurement function in Crofton's formula. Advances in Applied Mathematics. 116:Article 102004. https://doi.org/10.1016/j.aam.2020.102004

Eriksen, Rikke and Markus Kiderlen. "Uniqueness of the measurement function in Crofton's formula". *Advances in Applied Mathematics*. 2020. 116. https://doi.org/10.1016/j.aam.2020.102004

Eriksen R, Kiderlen M. Uniqueness of the measurement function in Crofton's formula. Advances in Applied Mathematics. 2020 May;116. 102004. https://doi.org/10.1016/j.aam.2020.102004

Eriksen, Rikke ; Kiderlen, Markus. / **Uniqueness of the measurement function in Crofton's formula**. In: Advances in Applied Mathematics. 2020 ; Vol. 116.

@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",

}

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

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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

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