On reduced polytopes

Abstract

A convex body $R$ in ${\mathbb{R}}^d$ is called reduced if the minimal width $\Delta (R^{\prime })$ of each convex body $R^{\prime }\subset R$ different from $R$ is strictly smaller than the minimal width $\Delta (R)$ of $R$. In this article we construct a reduced polytope in ${\mathbb{R}}^3$, i.e. we answer the following question posed by Lassak: do there exist reduced polytopes in ${\mathbb{R}}^d$, $d⩾3$? Also, we prove some properties of reduced polytopes in ${\mathbb{R}}^3$.

arXiv commentThis paper has been withdrawn by the author. This paper has been superseded by arXiv:1701.08629 (merged from arXiv:1605.06791 and arXiv:1607.08125)