Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 51, No. 1, pp. 237249 (2010) 

Convex sets with homothetic projectionsValeriu SoltanDepartment of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, VA 22030, USA, email: vsoltan@gmu.eduAbstract: Nonempty sets $X_1$ and $X_2$ in the Euclidean space $\R^n$ are called \textit{homothetic} provided $X_1 = z + \lambda X_2$ for a suitable point $z \in \R^n$ and a scalar $\lambda \ne 0$, not necessarily positive. Extending results of Süss and Hadwiger (proved by them for the case of convex bodies and positive $\lambda$), we show that compact (respectively, closed) convex sets $K_1$ and $K_2$ in $\R^n$ are homothetic provided for any given integer $m$, $2 \le m \le n  1$ (respectively, $3\le m\le n  1$), the orthogonal projections of $K_1$ and $K_2$ on every $m$dimensional plane of $\R^n$ are homothetic, where the homothety ratio may depend on the projection plane. The proof uses a refined version of Straszewicz's theorem on exposed points of compact convex sets. Keywords: antipodality, convex set, exposed points, homothety, linefree set, projection Classification (MSC2000): 52A20 Full text of the article:
Electronic version published on: 27 Jan 2010. This page was last modified: 28 Jan 2013.
© 2010 Heldermann Verlag
