# A concavity property for the measure of product sets in groups

Fundamenta Mathematicae (1992)

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

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topRuzsa, 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

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

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