Let $B_n$ be the $n$-dimensional unit ball, and $B_n(\varepsilon)$ be the spherical cap with height $\varepsilon$ I am interested in the quantity $$\Gamma:=\sup_{\Delta:\textrm{ inscribed simplex in }B_n(\varepsilon)}\mathrm{vol}(\Delta)$$ I think this should be a classical result and have very nice upper and lower bounds on the order $\varepsilon^{(n+1)/2}$. The thing I am interested in knowing some good bounds for the constants in terms of $n$, but unfortunately I wasn't able to settle down with a good reference for this. Any help would be greatly appreciated.

  • $\begingroup$ Do you know if the max volume simplex has a face flush with the base of the cap? $\endgroup$ – Joseph O'Rourke Feb 25 '15 at 12:34
  • 1
    $\begingroup$ @JosephO'Rourke see my answer below for a calculation if you assume this is true $\endgroup$ – Moritz Firsching Feb 21 '16 at 14:25

I don't know any reference for this, and I don't know if this should be a "classical result", but let me give a lower bound, which might even be tight.

Let's denote the base of the cap by $BS$. It is a sphere of dimension $n-2$ with radius $$r=\sqrt{1-(1-\varepsilon)^2}=\sqrt{2\varepsilon-\varepsilon^2}.$$

Let $\Delta_{n-1}^R$ denote a regular simplex of dimension $n-1$ inscribed in a sphere of dimension $n-2$ with radius $R$. If I am not mistaken, the volume can be calculated as follows:

$$\text{vol}_{n-1}(\Delta_{n-1}^1)=\frac{n^\frac{n}{2}}{(n-1)^{\frac{n-1}{2}}(n-1)!}$$ and $$\text{vol}_{n-1}(\Delta_{n-1}^R)=R^{n-1}\text{vol}_{n-1}(\Delta_{n-1}^1).$$

Now if we define an $n$-simplex $P_n(\varepsilon)$ as the convex hull of the apex of the cap together with the vertices of a regular $(n-1)$-simplex $\Delta_{n-1}^r$ inside the sphere $\partial BS$, we can calculate its $n$-dimensional volume as follows: $$\begin{align}\text{vol}_n(P_n(\varepsilon))&=\frac{\varepsilon}{n}\text{vol}_{n-1}(\Delta_{n-1}^r)\\ &=\frac{\varepsilon}{n} r^{n-1}\text{vol}_{n-1}(\Delta_{n-1}^1)\\ &=\frac{\varepsilon}{n} (\sqrt{2\varepsilon-\varepsilon^2})^{n-1}\text{vol}_{n-1}(\Delta_{n-1}^1)\\ &=\frac{\varepsilon}{n} (\sqrt{2\varepsilon-\varepsilon^2})^{n-1}\frac{n^\frac{n}{2}}{(n-1)^{\frac{n-1}{2}}(n-1)!} \end{align} $$

Here is an illustration for $n=3$: enter image description here This agrees with the order $\varepsilon^\frac{n+1}{2}$ for $\varepsilon\rightarrow 0$ that you expected and you can easily get complete asymptotics to all orders. Clearly this is a lower bound: $$\Gamma\geq\text{vol}_n(P_n(\varepsilon)).$$ I find it plausible that this bound is tight. For this you would need to show two things are true for small $\varepsilon$:

  • a largest simplex in the cap has all but one vertex in the base of the cap (see comment by Joseph O'Rourke.)
  • for a largest simplex those vertices in the base of the cap form a regular simplex.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.