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.

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.