On Young hulls of convex curves in \(\mathbb{R}^{2n}\) (Q1281744)

For a convex curve \(\gamma:S^1\to\mathbb R^{2n}\) the authors introduce a series of convex domains \(\mathcal Y\mathcal H_\gamma(\mu)\) called Young hulls. A special ruled hypersurface bounding \(\mathcal Y\mathcal H_\gamma(\mu)\) is found and the minimal convex subset spanning \(\mathcal Y\mathcal H_\gamma(\mu)\) and its dual as its convex hull are described. An integral formula for the Euclidean volume \(\text{Vol} \mathcal Y\mathcal H_\gamma(n)\) of the biggest Young hull called elliptic hull is presented. The volume of the elliptic hull for the generalized normalized ellipse in \(\mathbb R^4\) is explicitly calculated.