文摘
A random spherical polytope m>Pm>m>nm> in a spherically convex set pan data-equation-construct="true" class="math-equation-construct">pan data-equation-image="true" class="math-equation-image">pan>pan data-equation-mathml="true" class="math-equation-mathml" style="display:none"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow xmlns:w="http://www.wiley.com/namespaces/wiley" xmlns:wiley="http://www.wiley.com/namespaces/wiley/wiley" xmlns:cr="urn://wiley-online-library/content/render" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Kmi><mo>⊂mo><msup><mi mathvariant="double-struck">Smi><mi>dmi>msup>mrow>mml:math>pan>pan> as considered here is the spherical convex hull of m>nm> independent, uniformly distributed random points in m>Km>. The behaviour of m>Pm>m>nm> for a spherically convex set m>Km> contained in an open halfsphere is quite similar to that of a similarly generated random convex polytope in a Euclidean space, but the case when m>Km> is a halfsphere is different. This is what we investigate here, establishing the asymptotic behaviour, as m>nm> tends to infinity, of the expectation of several characteristics of m>Pm>m>nm>, such as facet and vertex number, volume and surface area. For the Hausdorff distance from the halfsphere, we obtain also some almost sure asymptotic estimates.
