Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 45, No. 2, pp. 549-555 (2004)

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Two optimization problems for convex bodies in the $n$-dimensional space

María A. Hernández Cifre, Guillermo Salinas and Salvador Segura Gomis

Dpto. de Matemáticas, Universidad de Murcia, 30100-Murcia, Spain, e-mail:; Dpto. Estadística y Matemática Aplicada, Universidad Miguel Hernández, 03202-Elche, Spain, e-mail:, Dpto. de Análisis Matemático, Universidad de Alicante, 03080-Alicante, Spain, e-mail:

Abstract: If $K$ is a convex body in the Euclidean space $E^n$, we consider the six classic geometric functionals associated with $K$: its $n$-dimensional volume $V$, ($n-1$)-dimensional surface area $F$, diameter $d$, minimal width $\omega$, circumradius $R$ and inradius $r$. We prove that the $n$-spherical symmetric slices are the convex bodies that maximize both, the volume and the surface area, when another two geometric magnitudes are fixed, specifically, for given values of the pairs of magnitudes ($\omega$, $d$) and ($\omega$, $R$). Besides, it is proved that the sets of constant width maximize the minimal width when the circumradius and the inradius are prescribed.

Classification (MSC2000): 52A20, 52A40

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Electronic version published on: 9 Sep 2004. This page was last modified: 4 May 2006.

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