Geodesic
Convex Hypersurfaces with Hyperplanar Intersections of Their Homothetic Copies
Journal of convex analysis, Tome 22 (2015) no. 1, pp. 145-159
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Extending a well-known characteristic property of ellipsoids, we describe all convex solids $K \subset \mathbb{R}^n$, possibly unboun\-ded, with the following property: for any vector $z \in \mathbb{R}^n$ and any scalar $\lambda \ne 0$ such that $K \ne z + \lambda K$, the intersection of the boundaries of $K$ and $z + \lambda K$ lies in a hyperplane. This property is related to hyperplanarity of shadow-boundaries of $K$ and central symmetricity of small 2-dimensional sections of $K$.
Classification : 52A20
Mots-clés : Besicovitch, body, convex, ellipse, ellipsoid, convex, quadric, section, shadow-boundary, solid 
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     author = {V. Soltan},
     title = {Convex {Hypersurfaces} with {Hyperplanar} {Intersections} of {Their} {Homothetic} {Copies}},
     journal = {Journal of convex analysis},
     pages = {145--159},
     year = {2015},
     volume = {22},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JCA_2015_22_1_JCA_2015_22_1_a7/}
}
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V. Soltan. Convex Hypersurfaces with Hyperplanar Intersections of Their Homothetic Copies. Journal of convex analysis, Tome 22 (2015) no. 1, pp. 145-159. http://geodesic.mathdoc.fr/item/JCA_2015_22_1_JCA_2015_22_1_a7/