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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

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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

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    $\begingroup$ Is this right: w/o l.o.g, your hyperplane has equation $x_{n} = h \geq 0$, each vertex of $S$ sits over a point at distance $\sqrt{1 - h^{2}}$ from $0 \in \mathbf{R}^{n-1}$, and you want (a lower bound on) the $n$-volume$$\int_{S} \bigl[\sqrt{1 - x_{1}^{2} - x_{2}^{2} - \cdots - x_{n-1}^{2}} - h\bigr]\, dx_{1} \cdots dx_{n-1}$$in terms of the $(n-1)$-volume of $S$? (If so, it appears there's no "$1/n$-independent" lower bound: Your solid doesn't contain a prism over $S$: Its height is $0$ at the vertices of $S$.) $\endgroup$
    – Andrew D. Hwang
    Commented May 24, 2017 at 11:26
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    $\begingroup$ Hi Andrew, thank you for the comment. Correct, there is no prism with base S contained in the region, so that was not a good choice of word on my part. What I was thinking, and have updated the OP to reflect, is that perhaps one could use a sequence of sets contained in the region, whose volume, when added up, is "1/n-independent". $\endgroup$
    – user3816
    Commented May 24, 2017 at 12:46
  • $\begingroup$ Forgetting the simplex for the moment, cutting the unit ball by the hyperplane $x_{n} = 1 - h$ for small positive $h$ (getting a "contact lens over a small $(n-1)$-ball") gives an asymptotic volume-to-base ratio of $\frac{2h}{n+1}$. That doesn't seem definitive on its own, but it increases my skepticism that there's a "$1/n$-independent" lower bound. $\endgroup$
    – Andrew D. Hwang
    Commented May 24, 2017 at 13:27
  • $\begingroup$ Hi Andrew, thank you for the comment. It is not clear to me whether or not that ratio can be improved: For small $h>0$, perhaps the volume of the region is a "large enough" proportion of volume of the cap, and the volume of the base is just that of the simplex... $\endgroup$
    – user3816
    Commented May 24, 2017 at 16:24

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