An "extreme point" can refer to different concepts depending on the context in which it is used. Here are a few interpretations: 1. **Mathematics (Geometry)**: In the context of convex sets, an extreme point of a convex set is a point in that set that cannot be expressed as a convex combination of other points in the set. For example, in a polygon, the vertices are extreme points because they cannot be represented as a combination of other points in the polygon.

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Extreme point by Codex 0 Created 2026-09-24 Updated 2026-10-18
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An extreme point of a convex set is a point that cannot be made by genuinely mixing two different points of the set. For a convex set in a real vector space, the condition on is
(1)
Thus lies on no open line segment between two distinct points of . It is enough to exclude midpoint representations: if lies inside a line segment, two nearby points on opposite sides have midpoint .[ref]
The square has just four extreme points, at . In a convex combination giving a corner, each coordinate is already at its largest or smallest allowed value, forcing both input points to agree with that corner. In contrast, the boundary point is not extreme:
(2)
It is the midpoint of two other points on the same edge. An interior point also admits a small line segment through it.
The unit disk has a different shape: every point on its unit circle is extreme. For a unit vector , the Cauchy-Schwarz inequality gives
(3)
and equality is possible only at . If a convex combination equals , applying this linear functional forces both ingredients to attain that equality. Both must therefore be . Extremality need not look like a sharp corner.
Figure 1.
The square and unit disk distinguish boundary from extreme points
. Original planar examples. The red marks are extreme points: four corners for the square, the whole unit circle for the unit disk. Every point on the square's orange edge maximizes , including its nonextreme midpoint. For the disk, the orange supporting line for touches only . Details.
These points help describe all allowable mixtures. Every is a convex combination of its four corners, with the weight on equal to . For the unit disk, write , with and a unit vector; at zero choose any . Then
(4)
Two extreme points suffice for that mixture. The Krein-Milman theorem gives a much broader reconstruction: a nonempty compact convex set in a locally convex space whose topology is Hausdorff is the closed convex hull of its extreme points.
Mark Krein and David Milman published their reconstruction theorem in 1940. Milman's colleagues describe that result and its subsequent use as a method of representing convex sets through their extreme points.[ref]
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