Abstract
It is remarkable that there is a duality in geometric tomography between results on projections of convex bodies and results on sections of star (rather than convex) bodies. The radial Blaschke addition, which is the dual version of Blaschke addition, as an operation between central symmetric star bodies is introduced in this paper. The relationship between it and the classical radial addition, many properties of radial Blaschke addition and related inequalities are established.
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References
Blaschke, W.: Kreis und Kugel, 2nd edn. W. de Gruyter, Berlin (1956)
Böröczky, K.J., Schneider, R.: Stable determination of convex bodies from sections. Studia Sci. Math. Hung. 46, 367–376 (2009)
Fenchel, W., Jessen, B.: Mengenfunktionen und konvexe Körper. Danske Vid. Selsk. Math.-Fys. Medd 16(3), 31 (1938)
Gardner, R.: Geometric Tomography. Encyclopedia of Mathematics and its Applications, vol. 58, 2nd edn. Cambridge University Press, Cambridge (2006)
Gardner, R.J., Hug, D., Weil, W.: Operations between sets in geometry. J. Eur. Math. Soc. 15, 2297–2352 (2013)
Gardner, R.J., Parapatits, L., Schuster, F.E.: A characterization of Blaschke addition. Adv. Math. 254, 396–418 (2014)
Groemer, H.: Geometric Applications of Fourier Series and Spherical Harmonics. Cambridge University Press, New York (1996)
Guo, L., Leng, G.: Determination of star bodies from p-centroid bodies. Proc. Indian Acad. Sci. Math. Sci. 123, 577–586 (2013)
Guo, L., Leng, G.: Stable determination of convex bodies from centroid bodies. Houston J. Math. 40, 395–406 (2014)
Guo, L., Leng, G., Lin, Y.: A stability result for \(p\)-centroid bodies. Bull. Korean Math. Soc. 55, 139–148 (2018)
Koldobsky, A.: Fourier Analysis in Convex Geometry, Math. Surveys Monogr. Amer. Math. Soc., Providence (2005)
Minkowski, H.: Allgemeine Lehrsätze über die convexen Polyeder, Nachr. Ges. Wiss. Göttingen, 198–219, (1897). Gesammelte Abhandlungen, vol. II, pp. 103–121. Teubner, Leipzig, (1911)
Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia Math. Appl., vol. 151, expanded edn. Cambridge University Press, Cambridge (2014)
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Supported in part by the National Natural Science Foundation of China (Grant No. 11801151) and Technology Key Project of the Education Department of Henan Province (19A110022).
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Guo, L., Jia, H. The Dual Blaschke Addition. J Geom Anal 30, 3026–3034 (2020). https://doi.org/10.1007/s12220-019-00190-7
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DOI: https://doi.org/10.1007/s12220-019-00190-7