Abstract
Associated with the L p -curvature image defined by Lutwak, some inequalities for extended mixed p-affine surface areas of convex bodies and the support functions of L p -projection bodies are established. As a natural extension of a result due to Lutwak, an L p -type affine isoperimetric inequality, whose special cases are L p -Busemann-Petty centroid inequality and L p -affine projection inequality, respectively, is established. Some L p -mixed volume inequalities involving L p -projection bodies are also established.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Campi S and Gronchi P, The L p-Busemann-Petty centroid inequality, Adv. Math. 167 (2002) 128–141
Federer H, Geometric Measure Theory (New York: Springer-Verlag) (1969)
Firey W J, p-means of convex bodies, Math. Scand. 10 (1962) 17–24
Hardy G H, Littlewood J E and Pólya G, Inequalities (Cambridge: Cambridge Univ. Press) (1934)
Leichtweiß K, Bemerkungen zur Definition einer erweiterten Affinoberfläche von E. Lutwak, Manuscripta Math. 65 (1989) 181–197
Lutwak E, On some affine isoperimetric inequalities, J. Differential Geom. 23 (1986) 1–13
Lutwak E, Centroid bodies and dual mixed volumes, Proc. London Math. Soc. 60 (1990) 365–391
Lutwak E, Extended affine surface area, Adv. Math. 85 (1991) 39–68
Lutwak E, The Brunn-Minkowski-Firey theory I: Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993) 131–150
Lutwak E, The Brunn-Minkowski-Firey theory II: Affine and geominimal surface areas, Adv. Math. 118 (1996) 244–294
Lutwak E, Yang D and Zhang G, L p affine isoperimetric inequalities, J. Differential Geom. 56 (2000) 111–132
Petty C M, Projection bodies, in: Proc. Coll. Convexity, Coperhagen, 1965 (Coperhagen: Københavns Univ. Mat. Inst.) (1967) pp. 234–241
Petty C M, Isoperimetric problems, in: Proc. Conference Convexity and Combinatorial Geometry, Univ. of Oklahoma, 1971 (University of Oklahoma) (1972) pp. 26–41
Petty C M, Affine isoperimetric problems, Ann. N. Y. Acad. Sci. 440 (1985) 113–127
Petty C M, Geominimal surface area, Geom. Dedicata 3 (1974) 77–97
Schneider R, Zu einem problem von Shephard über die projektionen konvexer Körper, Math. Z. 101 (1967) 71–82
Schneider R and Weil W, Zonoids and related topics, in Convexity and its Applications (eds) P M Gruber and J M Wills (Basel: Birkhäuser) (1983)
Zhang G, Centered bodies and dual mixed volumes, Trans. Am. Math. Soc. 345 (1994) 777–801
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lv, S., Leng, G. The L p -curvature images of convex bodies and L p -projection bodies. Proc Math Sci 118, 413–424 (2008). https://doi.org/10.1007/s12044-008-0032-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12044-008-0032-6