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MSC: 52A15, 53C45, 53C65
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integral geometry; convex body; zonoid; support function
It is known that a local equatorial characterization of zonoids does not exist. The question arises: Is there a subclass of zonoids admitting a local equatorial characterization. In this article a sufficient condition is found for a centrally symmetric convex body to be a zonoid. The condition has a local equatorial description. Using the condition one can define a subclass of zonoids admitting a local equatorial characterization. It is also proved that a convex body whose boundary is an ellipsoid belongs to the class.
[1] Aramyan, R. H.: Reconstruction of centrally symmetric convex bodies in ${\mathbb R}^n$. Bul. Acad. Ştiinţe Repub. Mold., Mat. 65 (2011), 28-32. MR 2849225
[2] Aramyan, R. H.: Measures in the space of planes and convex bodies. J. Contemp. Math. Anal., Armen. Acad. Sci. 47 78-85 (2012), translation from Izv. Nats. Akad. Nauk Armen., Mat. 47 19-30 Russian (2012). MR 3287915 | Zbl 1302.53082
[3] Goodey, P., Weil, W.: Zonoids and generalisations. Handbook of Convex Geometry, Vol. A, B North-Holland, Amsterdam 1297-1326 (1993), P. M. Gruber et al. 1297-1326. MR 1243010 | Zbl 0791.52006
[4] Leichtweiss, K.: Konvexe Mengen. Hochschulbücher für Mathematik 81 VEB Deutscher Verlag der Wissenschaften, Berlin (1980), German. MR 0586235 | Zbl 0442.52001
[5] Nazarov, F., Ryabogin, D., Zvavitch, A.: On the local equatorial characterization of zonoids and intersection bodies. Adv. Math. 217 (2008), 1368-1380. DOI 10.1016/j.aim.2007.08.013 | MR 2383902 | Zbl 1151.52002
[6] Panina, G. Yu.: Representation of an $n$-dimensional body in the form of a sum of $(n-1)$-dimensional bodies. Izv. Akad. Nauk Arm. SSR, Mat. 23 (1988), 385-395 Russian translation in Sov. J. Contemp. Math. Anal. 23 (1988), 91-103. MR 0997401 | Zbl 0679.52006
[7] Schneider, R.: Über eine Integralgleichung in der Theorie der konvexen Körper. Math. Nachr. 44 (1970), 55-75 German. DOI 10.1002/mana.19700440105 | MR 0275286 | Zbl 0162.54302
[8] Schneider, R.: Convex Bodies: the Brunn-Minkowski Theory. Encyclopedia of Mathematics and Its Applications 44 Cambridge University Press, Cambridge (1993). MR 1216521 | Zbl 0798.52001
[9] Schneider, R., Weil, W.: Zonoids and Related Topics. Convexity and Its Applications Birkhäuser, Basel (1983), 296-317. MR 0731116 | Zbl 0524.52002
[10] Weil, W.: Blaschkes Problem der lokalen Charakterisierung von Zonoiden. Arch. Math. 29 (1977), 655-659 German. DOI 10.1007/BF01220469 | MR 0513967 | Zbl 0382.52006
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