Consider the $n$-dimensional unit ball $B$ centered at the origin and a hyperplane $H$ that intersects $B$. Suppose that there is a simplex $S$ inscribed in $B\cap H$, so that the vertices of $S$ lie in the boundary of $B\cap H$. Let $|S|$ denote the $(n-1)$ volume of $S$. Is there a formula for the volume of the region of the (smaller) cap that lies above $S$, in terms of $|S|$? e.g. volume of $S$ times an integral, etc.

The picture looks like this in three dimensions, except instead of a facet on the top of the object, we have a spherical polytope, and instead of going directly up, we go radially out to the cap from the origin: https://math.stackexchange.com/questions/435060/volume-of-n-dimensional-solid-w-n-1-dimensional-simplex-as-a-base

If one cannot come up with such a formula, can one obtain a lower bound for the volume of this region, again in the form "$|S|\times$ something", that does not involve the factor $1/n$? In the link below, someone asked a similar question about inscribing an $n$-simplex in a cap, but the volume of this set involves a $1/n$ factor. Perhaps one could use a sequence of sets contained in the region, whose volume, when added up, gets rid of this factor?

Thank you in advance.

Related: Maximal volume of a simplex inscribed in a spherical cap