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The approximate Carath\'eodory theorem states that given a compact convex set $\mathcal{C}\subset\mathbb{R}^n$ and $p\in\left[2,+\infty\right[$, each point $x^*\in\mathcal{C}$ can be approximated to $\epsilon$-accuracy in the $\ell_p$-norm as the convex combination of $\mathcal{O}(pD_p^2/\epsilon^2)$ vertices of $\mathcal{C}$, where $D_p$ is the diameter of $\mathcal{C}$ in the $\ell_p$-norm.
Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen
C. Carathéodory · 1907
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