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Recently, I discovered a precise formulation directly relevant to my research:

  1. Let $X=[0,1]^n$. For all $n>2$ does $X$ admit a unique codimension one surface of revolution, $L$, with a complete metric (away from the cone points) and an embedding $e :L\hookrightarrow X$, which maximizes volume of $L$ while preserving constant positive sectional curvature? Let the cone points $p,q$ satisfy $\mathrm{sup~dist}_n(p,q)=\sqrt{n}$ where $p,q \in L$.
  1. This relates to another problem involving the volume and curvature of $L$: Does $\mathrm{Vol}(L)K=1$ for $n>2$?

What's interesting about these problems is the interplay between differential geometry concepts like curvature, volume, and optimization techniques with respect to those quantities and constraints. The closest reference I could find was this document which only really works with surfaces in $\mathbf R^3$.

Are there any other related works, ideally saying things about higher dimensions? The linked document provides the groundwork for a rigorous proof of the case $n=3$ for my first problem. It also gives evidence for the second problem, since we can obtain by numerical means, $K\approx 2$ and $\mathrm{Vol(L)}\approx1/2$.

At the end of the day I am interested in more resources or progress on the first problem I listed. The second is more computational. I am completely open to accepting an answer that rephrases the problem in a different way or connects to a different mathematical field.

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  • $\begingroup$ You mention a codimension one surface of revolution in a box $X$. What is the definition of surface of revolution? I would imagine that in $\mathbb{R}^n$, I would look at the orthogonal group $O(n-1)$ acting by fixing every point of some line in $\mathbb{R}^n$. A surface of revolution might be a codimension one submanifold invariant under $O(n-1)$. But for $X$, it surely matters which line we pick since the obvious line, say along the axis of one variable, intersects $X$ in a 1-dimensional corner of $X$. $\endgroup$
    – Ben McKay
    Commented Dec 4 at 13:30
  • $\begingroup$ @BenMcKay Yes it matters which line we pick, but the condition I gave: "The cone points $p,q$ satisfy $\mathrm{sup~dist}_n(p,q)=\sqrt{n}$ where $p,q \in L$," already tells us which line we should pick. In fact considering the space of all $p,q$ satisfying that distance criterion, we get $2^{n-1}$ surfaces of revolution (which all differ by some rotation, so we can consider just a single candidate out of these). I.e. the single candidate surface is unique up to some rotation. $\endgroup$
    – geocalc33
    Commented Dec 5 at 12:43

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