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Inequalities involving integrals of polar-conjugate concave functions. (English) Zbl 0903.52007

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 \(K\).

MSC:

52A40 Inequalities and extremum problems involving convexity in convex geometry
26B25 Convexity of real functions of several variables, generalizations

Citations:

Zbl 0737.52002
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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
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