Editing Arrow–Debreu model (section)

=== The role of convexity ===
[[Image:Unit circle.svg|thumb|right|alt=Picture of the unit circle|A quarter turn of the convex [[unit disk]] leaves the point&nbsp;''(0,0)'' fixed but moves every point on the non–convex [[unit circle]].]]
{{Main|Kakutani fixed-point theorem}}
{{See also|Convex set|Compact set|Continuous function|Fixed-point theorem|Brouwer fixed-point theorem}}

In 1954, [[Lionel McKenzie|McKenzie]] and the pair [[Kenneth Arrow|Arrow]] and [[Gérard Debreu|Debreu]] independently proved the existence of general equilibria by invoking the [[Kakutani fixed-point theorem]] on the [[fixed-point theorem|fixed point]]s of a [[hemicontinuity|continuous]] [[multivalued function|function]] from a [[compact space|compact]], convex set into itself<!-- Links to Kakutani's theorem are accurate, but the article description gives the simpler case Brouwer fixed point theorem for a point-valued function. -->. In the Arrow–Debreu approach, convexity is essential, because such fixed-point theorems are inapplicable to non-convex sets. For example, the rotation of the [[unit circle]] by&nbsp;90&nbsp;degrees lacks fixed points, although this rotation is a continuous transformation of a compact set into itself; although compact, the unit circle is non-convex. In contrast, the same rotation applied to the [[unit disk|convex hull of the unit circle]] leaves the point&nbsp;''(0,0)'' fixed. Notice that the Kakutani theorem does not assert that there exists exactly one fixed point. Reflecting the unit&nbsp;disk across the y-axis leaves a vertical segment fixed, so that this reflection has an infinite number of fixed points.