On planar convex bodies of given Minkowskian thickness and least possible area
Let \(B\) be the unit disc in a \(2\)-dimensional normed space (Minkowski space). The {Minkowski thickness} of a convex body \(K\), \(\Delta_B(K)\) is defined to be \(\min\{ (h_K(u) + h_K(-u))/h_B(u) : | u| = 1\}\) where \(h\) denotes the support function of the indicated body and \(| \cdot| \) is an auxiliary Euclidean norm. The author investigates the problem of minimizing the area of \(K\) subject to the condition \(\Delta_B(K) = 1\) and denotes the set of minimizers by \(\chi(B)\). In the Euclidean case (with \(B=E\)) it is known [\textit{J. Pál}, Math. Ann. 83, 311--319 (1921; JFM 48.0838.02)] that \(\chi(E)\) is the set of equilateral triangles of height \(1\). The author proves that for general \(B\), (i) \(\chi(B)\) always contains triangles and the difference body of any such triangle (an affine regular hexagon) is circumscribed to \(B\); (ii) if this set does not exhaust \(\chi(B)\) then the remaining elements are quadrilaterals; and (iii) he characterizes when quadrilaterals are present in \(\chi(B)\). The question in dimension \(\geq 3\) (even for the Euclidean case) is completely open.
scientific article; zbMATH DE number 2155977