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Every point in the convex hull of a subset of ${\mathbb{R}}^d$ is a convex combination of at most $d+1$ points of that subset. Thus in the plane three ingredients always suffice, however many samples were supplied. The representation of $p=(2,1)$ in convex hull illustrates this bound. Two ingredients need not suffice: the centroid of a triangle with only its three vertices as input is on none of their three pairwise line segments.

A short proof outline removes redundant ingredients. With more than $d+1$ points, the augmented vectors $(s_i,1)$ are linearly dependent. A dependence lets the weights vary while preserving both their sum and the represented point. Move along it until one positive weight first becomes zero; all weights stay nonnegative. Repeating leaves at most $d+1$ positive weights.

In his 1911 paper, Constantin Carathéodory used convex geometry to investigate the possible Fourier coefficients of positive harmonic functions. Its introduction interprets convex mixtures through the center of mass, connecting finite-dimensional geometry to constraints on a function's coefficients.^[ref]