Some geometric properties of closed space curves and convex bodies

Abstract:

The main results of the paper are as follows.

1. On each smooth closed oriented curve in $\mathbb {R}^n$, there exist two points the oriented tangents at which form an angle greater than $\pi /2+\sin ^{-1}\frac 1{n-1}$.

2. If $n$ is odd, then an $(n+1)$-gon with equal sides and lying in a hyperplane can be inscribed in each smooth closed Jordan curve in $\mathbb {R}^n$. In particular, a rhombus can be inscribed in each closed curve in $\mathbb {R}^3$.

3. A right prism with rhombic base and an arbitrary ratio of the base edge to the lateral edge can be inscribed in each smooth strictly convex body $K\subset \mathbb {R}^3$.