A concavity property for the measure of product sets in groups

Imre Ruzsa

Fundamenta Mathematicae (1992)

  • Volume: 140, Issue: 3, page 247-254
  • ISSN: 0016-2736

Abstract

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Let G be a connected locally compact group with a left invariant Haar measure μ. We prove that the function ξ(x) = inf μ̅(AB): μ(A) = x is concave for any fixed bounded set B ⊂ G. This is used to give a new proof of Kemperman’s inequality μ ̲ ( A B ) ≥ m i n ( μ ̲ ( A ) + μ ̲ ( B ) , μ ( G ) ) for unimodular G.

How to cite

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Ruzsa, Imre. "A concavity property for the measure of product sets in groups." Fundamenta Mathematicae 140.3 (1992): 247-254. <http://eudml.org/doc/211944>.

@article{Ruzsa1992,
abstract = {Let G be a connected locally compact group with a left invariant Haar measure μ. We prove that the function ξ(x) = inf μ̅(AB): μ(A) = x is concave for any fixed bounded set B ⊂ G. This is used to give a new proof of Kemperman’s inequality $μ̲(AB) ≥ min (μ̲(A) + μ̲(B), μ(G))$ for unimodular G.},
author = {Ruzsa, Imre},
journal = {Fundamenta Mathematicae},
keywords = {locally compact group; left Haar measure; inner and outer measures; impact function; continuous and concave},
language = {eng},
number = {3},
pages = {247-254},
title = {A concavity property for the measure of product sets in groups},
url = {http://eudml.org/doc/211944},
volume = {140},
year = {1992},
}

TY - JOUR
AU - Ruzsa, Imre
TI - A concavity property for the measure of product sets in groups
JO - Fundamenta Mathematicae
PY - 1992
VL - 140
IS - 3
SP - 247
EP - 254
AB - Let G be a connected locally compact group with a left invariant Haar measure μ. We prove that the function ξ(x) = inf μ̅(AB): μ(A) = x is concave for any fixed bounded set B ⊂ G. This is used to give a new proof of Kemperman’s inequality $μ̲(AB) ≥ min (μ̲(A) + μ̲(B), μ(G))$ for unimodular G.
LA - eng
KW - locally compact group; left Haar measure; inner and outer measures; impact function; continuous and concave
UR - http://eudml.org/doc/211944
ER -

References

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  1. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Springer, New York 1963. Zbl0115.10603
  2. Kemperman, On products of sets in a locally compact group, Fund. Math. 56 (1964), 51-68. Zbl0125.28901
  3. Kneser, Summenmengen in lokalkompakten abelschen Gruppen, Math. Z. 66 (1956), 88-110. 
  4. Macbeath, On measure of sum sets II. The sum-theorem for the torus, Proc. Cambridge Philos. Soc. 49 (1953), 40-43. Zbl0052.26301
  5. Plünnecke, Eigenschaften und Abschätzungen von Wirkungsfunktionen, Ges. Mathematik und Datenverarbeitung, Bonn 1969. 
  6. Raikov, On the addition of point sets in the sense of Schnirelmann, Mat. Sb. 5 (47) (1939), 425-440 (in Russian). Zbl0022.21003
  7. Shields, Sur la mesure d'une somme vectorielle, Fund. Math. 42 (1955), 57-60. Zbl0065.01702

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