Projection inequalities for antichains
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F20%3A00532203" target="_blank" >RIV/67985840:_____/20:00532203 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1007/s11856-020-2013-0" target="_blank" >https://doi.org/10.1007/s11856-020-2013-0</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s11856-020-2013-0" target="_blank" >10.1007/s11856-020-2013-0</a>
Alternative languages
Result language
angličtina
Original language name
Projection inequalities for antichains
Original language description
Let n be an integer with n ≥ 2. A set A ⊆ ℝn is called an antichain (resp. weak antichain) if it does not contain two distinct elements x = (x1, …, xn) and y = (y1, …, yn) satisfying xi ≤ yi (resp. xi < yi) for all i ∈ {1, …, n}. We show that the Hausdorff dimension of a weak antichain A in the n-dimensional unit cube [0, 1]n is at most n − 1 and that the (n − 1)-dimensional Hausdorff measure of A is at most n, which are the best possible bounds. This result is derived as a corollary of the following projection inequality, which may be of independent interest: The (n −1)- dimensional Hausdorff measure of a (weak) antichain A ⊆ [0, 1]n cannot exceed the sum of the (n − 1)-dimensional Hausdorff measures of the n orthogonal projections of A onto the facets of the unit n-cube containing the origin. For the proof of this result we establish a discrete variant of the projection inequality applicable to weak antichains in ℤn and combine it with ideas from geometric measure theory.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
<a href="/en/project/GJ18-01472Y" target="_blank" >GJ18-01472Y: Graph limits and inhomogeneous random graphs</a><br>
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2020
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Israel Journal of Mathematics
ISSN
0021-2172
e-ISSN
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Volume of the periodical
238
Issue of the periodical within the volume
1
Country of publishing house
IL - THE STATE OF ISRAEL
Number of pages
30
Pages from-to
61-90
UT code for WoS article
000534408400006
EID of the result in the Scopus database
2-s2.0-85085371997