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Journal of Convex Analysis 22 (2015), No. 1, 145--159
Copyright Heldermann Verlag 2015

Convex Hypersurfaces with Hyperplanar Intersections of Their Homothetic Copies

Valeriu Soltan
Dept. of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, VA 22030, U.S.A.


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 $K$.

Keywords: Besicovitch, body, convex, ellipse, ellipsoid, convex, quadric, section, shadow-boundary, solid.

MSC: 52A20

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