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

Summary

Given <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>r>1$, we search for the convex body of minimal volume in $\mathbb{E}^3$ that contains a unit ball, and whose extreme points are of distance at least $r$ from the centre of the unit ball. It is known that the extremal body is the regular octahedron and icosahedron for suitable values of $r$. In this paper we prove that if $r$ is close to one then the typical faces of the extremal body are asymptotically regular triangles. In addition we prove the analogous statement for the extremal bodies with respect to the surface area and the mean width.