Ruzsa, Imre Z. Diameter of sets and measure of sumsets. (English) Zbl 0737.28006 Monatsh. Math. 112, No. 4, 323-328 (1991). We define the diameter of a bounded set \(B\) as \(D(B)=\sup B-\inf B\). Let \(A\), \(B\) be bounded sets of reals with \(\underline\mu(A)=a\), \(\underline\mu(B)=b\), \(D(B)=D\) (\(\underline\mu\) denotes inner Lebesgue measure). We establish the inequality \[ \underline\mu(A+B)\geq\min (a+D,(\sqrt a+\sqrt{b/2})^ 2). \] In particular, if \(a\geq D^ 2/(2b)\), then \[ \underline\mu(A+B)\geq a+D.\tag{1} \] This improves the familiar inequality \(\underline\mu(A+B)\geq\underline\mu(A)+\underline\mu(B)\) for \(a\) not too small. A more complicated bound is also given that yields the best possible lower estimate of \(\underline\mu(A+B)\) in terms of \(a\), \(b\) and \(D\), and also gives the exact bound after which (1) holds. If \(A\) is a long interval, then there is equality in (1). Reviewer: I.Z.Ruzsa Cited in 6 Documents MSC: 28A75 Length, area, volume, other geometric measure theory 26B15 Integration of real functions of several variables: length, area, volume 11P99 Additive number theory; partitions Keywords:measure of sumsets; diameter of a bounded set PDFBibTeX XMLCite \textit{I. Z. Ruzsa}, Monatsh. Math. 112, No. 4, 323--328 (1991; Zbl 0737.28006) Full Text: DOI EuDML References: [1] Kemperman, J. H. B.: On products of sets in a locally compact group. Fundamenta Math.56, 51-68 (1964). · Zbl 0125.28901 [2] Raikov, D. A.: On the addition of point sets in the sense of Schnirelmann (in Russian). Mat. Sbornik5, 425-440 (1939). · Zbl 0022.21003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.