<div class="csl-bib-body">
<div class="csl-entry">Ludwig, M. (2024, June 5). <i>The Hadwger Theorem</i> [Presentation]. ISTA Mathematics Colloquium, Mara Gugging, Austria.</div>
</div>
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dc.identifier.uri
http://hdl.handle.net/20.500.12708/198013
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dc.description.abstract
In 1952, Hugo Hadwiger established a remarkable classification result. He proved that all rigid motion-invariant, continuous valuations on the space of convex bodies (compact convex sets) in $\R^n$ are linear combinations of intrinsic volumes. Here, a functional Z defined on convex bodies is called a valuation (or additive) if
\[\oZ(K)+\oZ(L)=\oZ(K\cup L) + \oZ(K\cap L) \]
for all convex bodies $K$ and $L$ such that $K\cup L$ is again a convex body. In $\R^3$, the Hadwiger theorem states that for every rigid motion-invariant, continuous valuation $\oZ$, there are $c_0, c_1, c_2, c_3\in \R$ such that
\[\oZ(K) = c_0 V_0(K) + c_1 V_1(K) + c_2 V_2(K) + c_3 V_3(K)\]
for every convex body in $\R^3$. Here, $V_0(K)=1$ (the Euler characteristic) and $V_3(K)$ is the 3-dimensional volume of K, while $V_2(K)$ is (up to a constant multiple) the perimeter of $K$ and $V_1(K)$ its mean width.
We will discuss this result, some of its consequences and applications, and some of the many results it inspired. In particular, we will describe a recent functional version of the Hadwiger theorem (joint work with Andrea Colesanti and Fabian Mussnig).
en
dc.description.sponsorship
FWF - Österr. Wissenschaftsfonds
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dc.language.iso
en
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dc.subject
convex body
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dc.subject
convex function
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dc.subject
valuation
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dc.title
The Hadwger Theorem
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dc.type
Presentation
en
dc.type
Vortrag
de
dc.relation.grantno
P 34446-N
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dc.type.category
Presentation
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tuw.publication.invited
invited
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tuw.project.title
Bewertungen auf konvexen Funktionen
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tuw.researchTopic.id
A3
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tuw.researchTopic.name
Fundamental Mathematics Research
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tuw.researchTopic.value
100
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tuw.publication.orgunit
E104-06 - Forschungsbereich Konvexe und Diskrete Geometrie
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tuw.author.orcid
0000-0002-7389-6720
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tuw.event.name
ISTA Mathematics Colloquium
en
tuw.event.startdate
05-06-2024
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tuw.event.enddate
05-06-2024
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tuw.event.online
On Site
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tuw.event.type
Event for scientific audience
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tuw.event.place
Mara Gugging
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tuw.event.country
AT
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tuw.event.institution
Institute of Science and Technology Austria
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tuw.event.presenter
Ludwig, Monika
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wb.sciencebranch
Mathematik
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wb.sciencebranch.oefos
1010
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wb.sciencebranch.value
100
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item.languageiso639-1
en
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item.openairetype
conference presentation
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item.grantfulltext
none
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item.fulltext
no Fulltext
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item.cerifentitytype
Publications
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item.openairecristype
http://purl.org/coar/resource_type/R60J-J5BD
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crisitem.author.dept
E104-06 - Forschungsbereich Konvexe und Diskrete Geometrie
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crisitem.author.orcid
0000-0002-7389-6720
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crisitem.author.parentorg
E104 - Institut für Diskrete Mathematik und Geometrie
