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I apologize for using non-common language. When this problem comes to my mind, it seems quite easy but It's not.

Maybe It can be rewritten as,

There exists a unique facet containing the most far points from one specific point as a face or a point of given regular polytope.

How do I prove this result under the theory of poyltope?

I'm not familiar with the theory of regular polytopes, so I think it's better to be recommended the reference for this.

EDIT: Actually, I can prove this for investigating every regular polytope. It's possible because I saw that there is only three types of regular polytopes in higher dimension. I need the proof from the definitions (and the results of theory) not by searching all cases.

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  • $\begingroup$ What symmetries can you prove from the definitions? If you can prove geometric but not isometric symmetries, it will be rough going. For a class of graphs embeddable in a cylinder I found some non-regular counterexamples to a similar sounding statement. Gerhard "Check Out Antipodes In Graphs" Paseman, 2019.10.23. $\endgroup$
    – Gerhard Paseman
    Commented Oct 23, 2019 at 15:02
  • $\begingroup$ @Gerhard. Thank you for your suggestion. That paper may help me. Anyway, my attention is in spherical regular polytopes. I use "the most far" in the sense of graph theory(that is, distance is the length of the shortest path between them). I won't consider the metric. $\endgroup$
    – ChoMedit
    Commented Oct 24, 2019 at 0:17

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