Two unsolved geometric problems (Q1781312)

It is known that any three points on \(S^2\) in \(\mathbb R^3\) can be placed by rotation of \(S^2\) into a level set of any prescribed continuous function \(f: S^2 \to \mathbb R\). It has been conjectured that all planar quadruples of points on \(S^2\) (i.e. quadruples of points lying on a circle) also possess this property. In Theorem 1.1, the author deals with an arbitrary quadruple \(x = (x^1, x^2, x^3, x^4)\) and establishes three alternatives two of which are rather intricate and the third one is just that \(x\) can be placed by a rotation of \(S^2\) into a level set of \(f\). Another topic of the paper are polygons inscribed in a smooth Jordan curve in \(\mathbb R^2.\) The author proves in particular the following (Theorem 2.1(1)). Each convex set \(K\) in \(\mathbb R^2\) is circumscribed to a mirror symmetric affine image of a regular pentagon. (By a convex set \(K\), the author actually means a compact convex body \(K\).)