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Let $X$ be composed of $d$ different vectors of $\mathbb{R}^n$ : $X=\{x_1,\ldots,x_d\}$ and $H$ be the convex hull of $X$. Each vector $y\in H$ can be expressed as $$y=\sum_{i=1}^d a_i x_i,$$ with non-negative weights $a_i$ and $\sum_{i=1}^da_i=1$.

My question is the following:

Given $y\in H$, is there a simple necessary and sufficient condition for the uniqueness of such a convex decomposition?

Such a characterization should both involve $y$ and $X$, but I really don't see how to proceed to find it.

TerranDrop
  • 840
  • thank you for pointing out my flawed reasoning. the situation is more complicated as can be seen in $\mathbb{R}^2$: consider any $n$-gon and the set of its triangulations. for each triangulation and for any given point $P$ inside the $n$-gon's convex hull there is exactly one triangle and thus 3 $n$-gon points who contain $P$. thus a convex decomposition need only use these 3 points. since there are at least 2 intersection-free triangulations for each convex $n$-gon ($n>3$), the decomposition cannot be unique. – collapsar Feb 25 '14 at 14:50
  • (in the above outline, i've ignored the special case of $P$ being located on the border of 2 tesselating triangles, which means that for this tesselation only two points are needed for the convex combination). possibly this generalizes to each convex point set $\mathcal{P} \in \mathbb{R^d}, |\mathcal{P}| > d$ and the tesselation of its convex hull into d-dimensional simplices. – collapsar Feb 25 '14 at 15:02

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