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Joseph O'Rourke
Joseph O'Rourke
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I seek a reference—not a proof—that if $P_1$ and $P_2$ are two convex polygons on a sphere composed of geodesic segments, contained in a hemisphere, and $P_1 \subseteq P_2$, then the perimeter of $P_1$ is $\le$ the perimeter of $P_2$. This is a spherical variant of the analogous claim in the plane, which was proved by Noam Elkies in an earlier MO question. I know how to prove the spherical version mimicking Noam's proof; the essence was used here.

         
I need to use this claim in a paper, and I'd prefer to cite a reference rather than include my own proof.

I seek a reference—not a proof—that if $P_1$ and $P_2$ are two convex polygons on a sphere composed of geodesic segments, and $P_1 \subseteq P_2$, then the perimeter of $P_1$ is $\le$ the perimeter of $P_2$. This is a spherical variant of the analogous claim in the plane, which was proved by Noam Elkies in an earlier MO question. I know how to prove the spherical version mimicking Noam's proof; the essence was used here.

         
I need to use this claim in a paper, and I'd prefer to cite a reference rather than include my own proof.

I seek a reference—not a proof—that if $P_1$ and $P_2$ are two convex polygons on a sphere composed of geodesic segments, contained in a hemisphere, and $P_1 \subseteq P_2$, then the perimeter of $P_1$ is $\le$ the perimeter of $P_2$. This is a spherical variant of the analogous claim in the plane, which was proved by Noam Elkies in an earlier MO question. I know how to prove the spherical version mimicking Noam's proof; the essence was used here.

         
I need to use this claim in a paper, and I'd prefer to cite a reference rather than include my own proof.
Source Link
Joseph O'Rourke
Joseph O'Rourke
  • 150.8k
  • 36
  • 358
  • 958

Perimeters of nested convex spherical polygons

I seek a reference—not a proof—that if $P_1$ and $P_2$ are two convex polygons on a sphere composed of geodesic segments, and $P_1 \subseteq P_2$, then the perimeter of $P_1$ is $\le$ the perimeter of $P_2$. This is a spherical variant of the analogous claim in the plane, which was proved by Noam Elkies in an earlier MO question. I know how to prove the spherical version mimicking Noam's proof; the essence was used here.

         
I need to use this claim in a paper, and I'd prefer to cite a reference rather than include my own proof.