Regular triangular pyramids held by a circle (Q1037209)

The paper is devoted to the problem of holding a convex body by a circle. This problem was first studied by \textit{T. Zamfirescu} [Geom. Dedicata 54, No.~3, 313--316 (1995; Zbl 0834.52003)] who showed that the convex bodies which cannot be held by a circle form a nowhere dense subset in the space of all convex bodies with the Hausdorff metric. The main result of the paper under review shows that a regular triangular pyramid which has height \(h\) and an equilateral triangle of side-length \(1\) as its base can be held by a circle if \(h>0.277\dots\).