Typical faces of extremal polytopes with respect to a thin three-dimensional shell (Q2456231)

For \(r> 1\), let \(\mathcal{F}_r\) be the family of convex bodies in \(E^3\) which contain the unit ball centered in \(o\) and whose extreme points are of distance at least \(r\) from \(o\). Moreover let \(P_r\in \mathcal{F}_r\) having minimal volume, \(Q_r\in \mathcal{F}_r\) having minimal surface area, and \(W_r\in \mathcal{F}_r\) having minimal mean width. The authors prove the following result: If \(r\) is close to one then all but at most \(c(r-1)^{\frac{1}{9}}\) percent of the boundaries of all of \(P_r, Q_r\) and \(W_r\) are the union of faces that are \((r-1)^{\frac{1}{9}}\)-close to the regular triangle of circumradius \(\sqrt{r^2-1}\), where \(c> 0\) is an absolute constant. Also, an element of \(\mathcal{F}_r\) that is close to be optimal with respect to all of volume, surface area and mean width is constructed.