Convex sets with homothetic projections

Valeriu Soltan

Abstract: 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, line-free set, projection

Classification (MSC2000): 52A20

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