Meyer, M.; Reisner, S. Inequalities involving integrals of polar-conjugate concave functions. (English) Zbl 0903.52007 Monatsh. Math. 125, No. 3, 219-227 (1998). An inequality of K. Mahler providing a lower bound for the product of the area of a plane convex figure and the area of its polar convex figure can be written in the form of integrals of polar-conjugate concave functions [see M. Meyer, Monatsh. Math. 112, No. 4, 297-301 (1991; Zbl 0737.52002)]. The authors generalise this inequality for integrals of powers of polar-conjugate functions and use it to derive further inequalities for the product of volumes of a convex body \(K\) and the polar to \(K-z\) where \(z\) is the Santaló point of \(K\). Reviewer: I.S.Molchanov (Glasgow) Cited in 3 ReviewsCited in 9 Documents MSC: 52A40 Inequalities and extremum problems involving convexity in convex geometry 26B25 Convexity of real functions of several variables, generalizations Keywords:concave function; polar set; inequality; polar-conjugate functions; volumes; convex body; Santaló point Citations:Zbl 0737.52002 PDFBibTeX XMLCite \textit{M. Meyer} and \textit{S. Reisner}, Monatsh. Math. 125, No. 3, 219--227 (1998; Zbl 0903.52007) Full Text: DOI EuDML References: [1] Alzer H (1992) On an integral inequality for concave functions. Acta Sci Math56: 79-82 · Zbl 0770.26007 [2] Mahler K (1939) Ein Minimalproblem f?r konvexe Polygone. Mathematica (Zutphen)B7: 118-127 · JFM 64.0732.01 [3] Meyer M (1991) Convex bodies with minimal volume product in ?2. Mh Math112: 297-301 · Zbl 0737.52002 · doi:10.1007/BF01351770 [4] Meyer M, Pajor A (1990) On the Blaschke-Santal? inequality, Arch Math55: 82-93 · Zbl 0718.52011 · doi:10.1007/BF01199119 [5] Petty CM (1985) Affine isoperimetric problems. In:Goodman JE, Lutwak E, Malkevitch J, Pollack R (eds) Discrete Geometry and Convexity. Ann New York Acad Sci440: 113-127 [6] Santal? LA (1949) Un invariante afin para los cuerpos convexos del espacio de n dimensiones. Portugaliae Math8: 155-161 [7] Schneider R (1993) Convex Bodies: The Brunn-Minkowski Theory, Cambridge: University Press · Zbl 0798.52001 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.