On quotients of ith affine surface areas

Following the volume difference function, we first introduce the notion of the affine surface area quotient function. We establish Brunn--Minkowski type inequalities for the affine surface area quotient function, which in special cases yield some well-known results.

On quotients of ith affine surface areas

Following the volume difference function, we first introduce the notion of the affine surface area quotient function. We establish Brunn--Minkowski type inequalities for the affine surface area quotient function, which in special cases yield some well-known results.

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  • Alesker, S., Bernig, A., Schuster, F.E.: Harmonic analysis of translation invariant valuations. Geom. Func. Anal. 21, 751–773 (2011).
  • Beckenbach, E.F., Bellman, R.: Inequalities. Berlin. Springer 1961.
  • Bonnesen, T., Fenchel, W.: Theorie der konvexen K¨ orper. Berlin. Springer 1934.
  • Dresher, M.: Moment spaces and inequalities. Duke Math. J. 20, 261–271 (1953).
  • Gardner, R.J.: Geometric Tomography. New York. Cambridge University Press 1996.
  • Leng, G.S.: The Brunn-Minkowski inequality for volume differences. Adv. Appl. Math. 32, 615–624 (2004).
  • Ludwig, M., Reitzner, M.: A classification of SL(n) invariant valuations. Ann. Math. 172, 1219–1267 (2010).
  • Lutwak, E.: Dual mixed volumes. Pacific J. Math. 58, 531–538 (1975).
  • Lutwak, E.: Centroid bodies and dual mixed volumes. Proc. London Math. Soc. 60, 365–391 (1990).
  • Lutwak, E.: Mixed affine surface area. J. Math. Anal. Appl. 125, 351–360 (1987).
  • Lutwak, E.: On some affine isoperimetric inequalities. J. Diff. Geom. 23, 1–13 (1986).
  • Lutwak, E.: The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas. Adv. Math. 118, 244–294 (1996).
  • Lv, S.: Dual Brunn-Minkowski inequality for volume differences. Geom. Dedicata 145, 169–180 (2010).
  • Parapatits, L., Schuster, F.E.: The Steiner formula for Minkowski valuations. Adv. Math. 230, 978–994 (2012).
  • Petty, C.M.: Geominimal surface area. Geom. Dedicata 3, 77–97 (1974).
  • Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory. Cambridge. Cambridge University Press 1993.
  • Schuster, F.E.: Crofton measures and Minkowski valuations. Duke Math. J. 154, 1–30 (2010).
  • Werner, E., Ye, D.: Inequalities for mixed p -affine surface area. Math. Ann. 347, 703–737 (2010).
  • Zhao, C.J., Leng, G.: Inequalities for dual quermassintegrals of mixed intersection bodies. Proc. Indian Math. Sci. (Math. Sci.) 115, 79–91 (2005).
  • Zhao, C.J., Leng, G., Debnath, L.: Some new Brunn-Minkowski-type inequalities in convex bodies. Inter. J. Math. Math. Sci. 2005, 895–915 (2005).
  • Zhao, C.J., Bencze, M.: The Aleksandrov-Fenchel type inequalities for volume differences. Balkan J. Geom. Appl. 15, 163–172 (2010).
  • Zhao, C.J.: Radial Blaschke-Minkowski homomorphisms and volume differences. Geom. Dedicata 154, 81–91 (2011). Zhao, C.J.: Lp-dual quermassintegral sums. Sci. China 50, 1347–1360 (2007).
  • Zhao, C.J, Bencze, M.: Lp-Minkowski and Aleksandrov-Fenchel type inequalities. Balkan J. Geom. Appl. 14, 128– 137 (2009).