Hunting for reduced polytopes (Q1991351)

A convex body \(K\) in Euclidean \(d\)-space \(\mathbb {R} ^d\) is called reduced if there is no convex body properly contained in \(K\) but having the same minimum width as \(K\). In this article, a reduced 3-polytope is constructed on twelve vertices in \(\mathbb {R} ^3\), which partially answers the following question posed three decades ago by \textit{M. Lassak} [Isr. J. Math. 70, No. 3, 365--379 (1990; Zbl 0707.52005)]: Given \(d \geq 3\), does there exist a reduced polytope in \(\mathbb {R} ^d\)? Interestingly, the constructed 3-polytope has the same symmetry as the Zalgaller-Johnson solid \(J_{84}\) [\textit{V. A. Zalgaller}, Semin. in Mathematics, V.A. Steklov Math. Inst., Leningrad 2, 95 p. (1969; Zbl 0177.24802)] although has a different combinatorial structure. The constructed 3-polytope is depicted in the article.