Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 42, No. 1, pp. 123-136 (2001)

Shaking Compact Sets

Stefano Campi, Andrea Colesanti, Paolo Gronchi

Dipartimento di Matematica Pura e Applicata "G. Vitali", Universita degli Studi di Modena, Via Campi 213/b, 41100 Modena, Italy, e-mail:; Dipartimento di Matematica "U. Dini", Universita degli Studi di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italy, e-mail:; Istituto di Analisi Globale ed Applicazioni, Consiglio Nazionale delle Ricerche, Via S. Marta 13/a, 50139 Firenze, Italy e-mail:

Abstract: \font\msbm=msbm10 \def\Rd{\hbox{\msbm R}^d} If $C$ is a compact subset of $\Rd$ and $H$ is a halfspace bounded by a hyperplane $\pi$, the set $\tilde C$ obtained by shaking $C$ on $\pi$ is defined as the set contained in $H$, such that for every line $\ell$ orthogonal to $\pi$, $\tilde C\cap \ell$ is a segment of the same length as $C\cap \ell$, and one of its endpoints is on $\pi$. It is shown that there exist $d+1$ hyperplanes such that every compact set can be reduced to a simplex, via repeated shaking processes on these hyperplanes.

Classification (MSC2000): 52A30

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