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  "body": "\n## Summary\n\nChoquet theory studies how a point of a compact [[convex set|02AS45T3H9]] is an average of its [[extreme points|3MSGFJM2NX]].\nThe average is taken with respect to a [[probability measure|059TJN0NZ1]] on the set, called a representing measure.\nIt refines the [[Krein–Milman theorem|RSRAY0Q68S]], which only says that the set is the closed convex hull of its extreme points.\nThe theory applies to sets of measures, of states and of positive functions, where extreme points are the simplest objects.\n\n## Explanations\n\n### L1 human\n\nIn a triangle, every point is a weighted average of the three corners, with unique weights.\nChoquet theory extends this picture to infinite-dimensional convex sets, where a point may need infinitely many extreme points and a measure replaces the finite weights.\nIn a square, the centre is the average of two opposite corners in two different ways.\nSo the weights need not be unique, and the sets with unique weights, the simplices, play a special role.\n",
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