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Given a point $p$ and an n-simplex $\triangle_n$, the problem of finding the point $p'$ which is inside $\triangle_n$ and nearest to $p$ is fairly straightforward if $\triangle_n$ is the standard simplex.

But what if $\triangle_n$ is an arbitrary simplex? It seems it wouldn't suffice to perform the projection in the space of the standard simplex and then apply a linear transform to return to the space of the arbitrary simplex, because then the projection to any face of $\triangle_n$ would in general be non-orthogonal.

What efficient techniques are known to project a point to an arbitrary n-simplex?

trbabb
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  • Maybe we can get a result like "the face on which $p$ will be projected on contains $p$'s nearest vertex of the simplex"? Then, we could use one of the well-known efficient nearest-point-search algorithm and iterate through all faces incident with the nearest point. – user7427029 Feb 21 '21 at 13:17
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    Use quadratic programming. – Rodrigo de Azevedo Feb 21 '21 at 14:38
  • Related: https://math.stackexchange.com/q/2292242/339790 and https://math.stackexchange.com/q/2210394/339790 and https://math.stackexchange.com/q/292804/339790 – Rodrigo de Azevedo Feb 21 '21 at 14:41
  • It looks like there are many techniques related to quadratic programming. Which is the preferred one for simplices? Is there a paper (or commonly known technique) I can refer/compare to? In other words, what do I select to evaluate whether my implementation is any good? – trbabb Mar 06 '21 at 05:36

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