Abstract
In this paper we shall generalize Shearer’s entropy inequality and its recent extensions by Madiman and Tetali, and shall apply projection inequalities to deduce extensions of some of the inequalities concerning sums of sets of integers proved recently by Gyarmati, Matolcsi and Ruzsa. We shall also discuss projection and entropy inequalities and their connections.
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G.R. Allan: An inequality involving product measures, in: Radical Banach Algebras and Automatic Continuity (J.M. Bachar et al., eds.), Lecture Notes in Mathematics 975, Springer-Verlag, 1981, 277–279.
N. Alon and M. Dubiner: Zero-sum sets of prescribed size, in: Combinatorics, Paul Erdős is eighty, Vol. 1, pp. 33–50, Bolyai Soc. Math. Stud., János Bolyai Math. Soc., Budapest, 1993.
P. Balister and J.P. Wheeler: The Erdős-Heilbronn problem for finite groups, Acta Arithmetica, 140 (2009), 105–118.
B. Bollobás and A. Thomason: Projections of bodies and hereditary properties of hypergraphs, Bull. London Math. Soc. 27 (1995), 417–424.
Yu.D. Burago and V.A. Zalgaller: Geometric Inequalities, Springer-Verlag, 1988, xiv+331pp.
F.R.K. Chung, R.L. Graham, P. Frankl and J.B. Shearer: Some intersection theorems for ordered sets and graphs, J. Combinatorial Theory A 43 (1986), 23–37.
H. Davenport: On the addition of residue classes, J. London Math. Soc. 10 (1935), 30–32.
A. Frank: Edge-connection of graphs, digraphs, and hypergraphs, in: More Sets, Graphs and Numbers, Bolyai Soc. Math. Stud. 15, Springer, Berlin, 2006, pp. 93–141.
K. Gyarmati, M. Matolcsi and I. Z. Ruzsa: Plünnecke's inequality for different summands, in: Building Bridges, Bolyai Soc. Mathematical Studies 19, ed. M. Grötschel, G. O. H. Katona, Springer-Bolyai 2008, 309–320.
K. Gyarmati, M. Matolcsi and I. Ruzsa: A superadditivity and submultiplicativity property for cardinalities of sumsets, Combinatorica 30 (2010), 163–174.
H. Hadwiger: Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer-Verlag, 1957, xiii+312pp.
T. S. Han: Nonnegative entropy measures of multivariate symmetric correlations, Information and Control 36 (1978), 133–156.
G. Károlyi: The Cauchy-Davenport theorem in group extensions, L'Enseignement Mathématique 51 (2005), 239–254.
L.H. Loomis and H. Whitney: An inequality related to the isoperimetric inequality, Bull. Amer. Math. Soc. 55 (1949), 961–962.
L. Lovász: Solution to Problem 11, see pp. 168–169 of’ Report on the 1968 Miklós Schweitzer Memorial Mathematical Competition’ (in Hungarian), Matematikai Lapok 20 (1969), 145–171.
L. Lovász: 2-matchings and 2-covers of hypergraphs, Acta Math. Acad. Sci. Hungar. 26 (1975), 433–444.
L. Lovász: On two minimax theorems in graph, J. Combinatorial Theory Ser. B 21 (1976), 96–103.
M. Madiman and P. Tetali: Sandwich bounds for joint entropy, Proc. IEEE Intl Symp. Inform. Theory, Nice, June, 2007.
M. Madiman and P. Tetali: Information inequalities for joint distributions, with interpretations and applications, IEEE Transactions on Information Theory 56 (2010), 2699–2713.
I. Z. Ruzsa: Cardinality questions about sumsets, in: Additive Combinatorics, CRM Proc. Lecture Notes 43, Amer. Math. Soc., Providence, RI, 2007, pp. 195–205.
J. P. Wheeler: The Cauchy-Davenport theorem for finite groups, preprint, http://www.msci.memphis.edu/preprint.html (2006).
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Research supported in part by NSF grants CCR-0225610, DMS-0505550 and W911NF-06-1-0076