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In mathematics, an extreme point of a convex set S in a real vector space is a point in S which does not lie in any open line segment joining two points of S. Intuitively, an extreme point is a "vertex" of S.
- The Krein–Milman theorem states that if S is convex and compact in a locally convex space, then S is the closed convex hull of its extreme points: In particular, such a set has extreme points.
The Krein–Milman theorem is stated for locally convex topological vector spaces. The next theorems are stated for Banach spaces with the Radon–Nikodym property:
- A theorem of Joram Lindenstrauss states that, in a Banach space with the Radon–Nikodym property, a closed and bounded set has an extreme point. (In infinite-dimensional spaces, the property of compactness is stronger than the joint properties of being closed and being bounded).
- A theorem of Gerald Edgar states that, in a Banach space with the Radon–Nikodym property, a closed and bounded set is the closed convex hull of its extreme points.
Edgar's theorem implies Lindenstrauss's theorem.
An extreme point or an extremal point is a point that belongs to the extremity of something. More specifically it may refer to:
- Extreme point in mathematics, a point in a convex set which does not lie in any open line segment joining two points in the set
- A point where some function attains its extremum
- A leaf vertex of a tree in graph theory
- Extreme points of Earth in geography, points of land that extend farther in one direction than any other part of that land
Usage examples of "extreme point".
The boat then skirted the shore, which, extending to the extreme point, hid all Tadorn's Fens.
The boat then skirted the shore, which, extending to the extreme point, hid all Tadorn’.
At Cape Momax the straits widened, and she was able to get round Narborough Isles and advance in a more southerly direction, till at length the rock of Cape Pilares, the extreme point of Desolation Island, came in sight, thirty-six hours after entering the straits.
At the extreme point of the Calabria, he touched with his spear a column on the sea-shore of Rhegium, ^32 proclaiming that ancient landmark to stand the immovable boundary of his kingdom.
Clouders were - depending who you talked to - sentient, semi-sentient, proto-sentient, a-sentient or just plain not remotely sentient -though that last extreme point of view tended to be held only by those for whom it would be convenient if it were true, such as those who could do useful, profitable things with a big cloud of gas.
He was a pedant, to the most extreme point, the greatest pedant I had met on earth, and with that had a vanity only befitting Alexander of Macedon.
From Lignitz, the extreme point of their western march, they turned aside to the invasion of Hungary.
Thus the exact measurement from the centre of the central altar to the extreme point of any one of the rounded rays would be three hundred feet (the width of the circle itself), or a total of six hundred feet from the rounded extremity of one ray or petal to the extremity of the opposite one.