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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]

Constantin Carathéodory was a mathematician whose work connected measure theory, complex analysis and the calculus of variations. He began his career as an engineer and studied mathematics while working on a dam in Egypt, before entering mathematical research. His later contributions included links between the calculus of variations and partial differential equations, as well as conformal maps and their boundary behavior.^[ref] His Caratheodory measurability criterion is a basic construction principle for measures: it selects those sets that split the outer measure of every test set additively, producing a domain on which countable additivity holds.

By Constantin Carathéodory. Rendiconti del Circolo Matematico di Palermo, volume 32, pp. 193–217 (1911).

Publisher metadata: link.springer.com/article/10.1007/BF03014795

English translation: nchlswllms.github.io/translations/cc1911.pdf

The translation is titled On the Domain of Variability of Fourier Coefficients of Positive Harmonic Functions and credits Nicholas J. Williams as translator. It records the original announcement on 26 March 1911 and printing on 16 August 1911; neither dates the translation. The original German scan was not accessible.

Cited passages: Introduction §4, original printed p. 195; §I opening.