The coordinate of the centroid of a polytope in n-dimension

Asked Oct 04 '19 at 16:58

Active Oct 04 '19 at 16:58

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Is it true that the coordinate of the mass center of a convex polytope (constructed by n vertices) in a n-dimensional space is actually the average of all the vertices' coordinates? If so, how to prove it?

For example, say we have a convex polytope with vertices $(x_{11}, x_{12}, ..., x_{1n}), (x_{21}, x_{22}, ..., x_{2n}), ..., (x_{n1}, x_{n2}, ..., x_{nn})$, is it true that its mass center (mass is uniformly distributed over the polytope) is at $\frac{(x_{11}+ x_{21}+ ... +x_{n1})}{n}, \frac{(x_{12}+ x_{22}+ ... +x_{n2})}{n}, ..., \frac{(x_{1n}+ x_{2n}+ ... +x_{nn})}{n}$?

Thanks!

asked Oct 04 '19 at 16:58

lebesgue

Is it true for a plane quadrilateral? – Angina Seng Oct 04 '19 at 17:11
i am not sure about that – lebesgue Oct 04 '19 at 17:56
Well, it may be worth experimenting... – Angina Seng Oct 04 '19 at 17:57
1

try a quad where two of the vertices are soooo close together that the quad looks almost like a triangle. – antkam Oct 04 '19 at 20:34

The coordinate of the centroid of a polytope in n-dimension

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