Planes and polytopes in n dimensional space

Asked Nov 09 '22 at 08:45

Active Nov 09 '22 at 08:45

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Imagine there are K vectors in n-dimensional space. I would like to:

validate whether they can correspond to K+1 planes enclosing a volume (is that called a polytope?)
validate whether a given further vector is completely inside this polytope, or not.

This should be known math problems I assume, but I do not know the names of corresponding solution approaches/terms.

asked Nov 09 '22 at 08:45

J. Doe

A re-formulation of your issue: "does the convex hull of these $K$ points have $K$ vertices (all points participate to the convex hull) ? " – Jean Marie Nov 09 '22 at 09:50
Are you sure that $K$ points define $K+1$ planes ? Take the case of the tretahedron (also called simplex) in $\mathbb{R}^3$ : its 4 vertices define 4 planes; not 5. Besides, you don't say if $K<n, K=n, K>n$... – Jean Marie Nov 09 '22 at 10:31
Any comment to my comments ? – Jean Marie Nov 09 '22 at 18:24
thank you - I think the term "convex hull" nails it. – J. Doe Nov 10 '22 at 04:21
You haven't answered my question about the tetrahedron (pyramid with triangular base) which has $K=4$ vertices and $K=4$ limiting planes – Jean Marie Nov 10 '22 at 17:53
right, then it's not K+1 – J. Doe Nov 10 '22 at 19:01

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