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Project an n-simplex of side length $a$ on it's ($n-1$)-dimensional circumsphere by a ray starting at the midpoint. Denote the images of the $n+1$ faces of dimension $n-1$ of the simplex by $A_1,\dots A_{n+1}$. Choose $a_i\in A_i, i=1,\dots, n+1$. Are there always $i, j\in \mathbb{N}$ with $d(a_i, a_j)>=a$?

Project an n-simplex of side length $a$ on it's ($n-1$)-dimensional circumsphere by a ray starting at the center. Denote the images of the $n+1$ faces of dimension $n-1$ of the simplex by $A_1,\dots A_{n+1}$. Choose $a_i\in A_i, i=1,\dots, n+1$. Are there always $i, j\in \mathbb{N}$ with $d(a_i, a_j)>=a$?

Project an n-simplex of side length a on it's ($n-1$)-dimensional circumsphere by a ray starting at the midpoint. Denote the images of the $n+1$ faces of dimension $n-1$ of the simplex by $A_1,\dots A_{n+1}$. Choose $a_i\in A_i, i=1,\dots, n+1$. Are there always $i, j\in \mathbb{N}$ with $d(a_i, a_j)>=a$?

Project an n-simplex of side length $a$ on it's ($n-1$)-dimensional circumsphere by a ray starting at the midpoint. Denote the images of the $n+1$ faces of dimension $n-1$ of the simplex by $A_1,\dots A_{n+1}$. Choose $a_i\in A_i, i=1,\dots, n+1$. Are there always $i, j\in \mathbb{N}$ with $d(a_i, a_j)>=a$?