Real-world problem:
Given $m$ registered 3D geometry meshes with some known consistent measurements (e.g. human bodies with height, hip girth, inseam etc.), I want to produce a new mesh for some other given measurements. Assuming that the new measurements lie within the convex hull of the previously known ones, the new mesh's vertices can be approximated as a convex combination of the old ones'. It is also desirable for this combination to be sparse.
Formal definition:
More formally, given a dictionary $D^{m \times n}$ of $n$ basis vectors (e.g. given body measurements) of size $m$ and an input vector $x^{m \times 1}$ (e.g. new body measurements), find a vector $\alpha^{n \times 1}$ that solves the following constrained optimization problem (for some constant L):
$$ \min_{\alpha} ||x-D\alpha||_2^2\\ s.t.\\ \sum_{i=1..m}\alpha_i=1\ \ and\ \ \forall{i}:\alpha_i\ge0\\ ||\alpha||_0 \le L $$
It seems like you mean $\alpha$ is $n \times 1$, since this $\alpha$ is the coefficients for your existing points, and you have $n$ points, whereas $m$ is the dimension. If $x$ lies in the convex hull of your points, then it is a theorem that you can find $L = m + 1$ points (or fewer) which have non-zero coefficients in $\alpha$ that solves your problem with best possible objective of $0$ (exact fit). Also, if the affine span of your points is full dimensional (dimension $m$), then you will always need $m + 1$ points to get a perfect solution if $x$ is in the interior of the convex hull.
Your convex hull is what is called a polytope, and you can triangulate a polytope into "simplexes" (higher dimensional triangles, e.g. tetrahedra in 3d, and in general full dimensional polytopes with $m+1$ vertices). The triangulation is a decomposition of the polytope into a disjoint union of simplexes where all vertices of simplexes are vertices of the polytope. So if you triangulate your polytope, then for any point $x$ in the convex hull there will be a unique simplex that contains $x$, and this will give you your $m+1$ points to use in $\alpha$. This is especially efficient say if $m \leq 6$ and $n$ is not too large (like $n \leq 1000$ if $m \leq 4$ and $n \leq 200$ if $5 \leq m \leq 6$).
There is free software for computing triangulations available online. The only downside is that you'll have to do some programming work to use the triangulation and figure out which simplex contains a given point $x$.
UPDATE: If the dimension is larger or you have a lot of points, there is a fairly fast polynomial time algorithm which uses linear programming (for the case $L = m + 1$). Simply solve for $\alpha$ with one point removed using linear programming feasibility, and if you get a solution then you know you can safely exclude that point. Otherwise you have to keep the point. Keep repeating until you get $m + 1$ points that have to be in the set of points to get a solution. Then restricting to this set of $m + 1$ points and solving one more linear programming feasibility problem will give you your solution. Worst case is you solve around $n$ linear programming feasibility problems.
If you want to set $L < m + 1$ then I think the problem is much harder.