Projection inequalities for antichains

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Projection inequalities for antichains

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Projection inequalities for antichains

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GJ18-01472Y" target="_blank" >GJ18-01472Y: Limity grafů a nehomogenní náhodné grafy</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2020

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Israel Journal of Mathematics

ISSN

0021-2172

e-ISSN

—

Svazek periodika

238

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

IL - Stát Izrael

Počet stran výsledku

30

Strana od-do

61-90

Kód UT WoS článku

000534408400006

EID výsledku v databázi Scopus

2-s2.0-85085371997