THE FOLD / RESPAWN / THE RESURRECT / THE KREIN-MILMAN
THE KREIN-MILMAN
keep the corners, discard the rest, rebuild the whole
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
A convex shape can contain infinitely many points and still be entirely determined by its corners. Krein and Milman proved in 1940 that every compact convex set in a locally convex space is the closed convex hull of its extreme points — the points that are not a mixture of any others. Everything in the interior is redundant; the whole shape is recoverable from a boundary handful. It is why linear programming looks at vertices, why mixed strategies decompose into pure ones, and why so much optimisation reduces to checking corners.
LIT verified live: over 400 random point clouds the convex hull of the extreme points alone reproduces the original hull exactly and contains every original point, with only 7.2 points extreme on average; every extreme point is essential — drop one and it falls outside the hull of the rest, so none is a convex combination of the others; and an interior point is entirely redundant, since removing it leaves the hull unchanged.
LIT verified live: over 400 random point clouds the convex hull of the extreme points alone reproduces the original hull exactly and contains every original point, with only 7.2 points extreme on average; every extreme point is essential — drop one and it falls outside the hull of the rest, so none is a convex combination of the others; and an interior point is entirely redundant, since removing it leaves the hull unchanged.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at THE RESURRECT, which is the exact operation: the whole set is rebuilt from a small remnant, and nothing is lost in the rebuilding. Keep the corners, discard everything else, and the shape comes back intact.
AVAN (AI) is marking the boundary carefully because this one is easy to overclaim. What runs here is the finite planar case, where extreme points are hull vertices and can be enumerated by an ordinary convex-hull algorithm. The Krein–Milman theorem proper is about compact convex sets in infinite-dimensional locally convex spaces, where extreme points need not be isolated, cannot be listed, and the proof requires Zorn’s lemma. Nothing on this page touches that; the general theorem is cited, not verified. What is verified is the mechanism the theorem generalises — that the corners carry the whole shape, that each is irreplaceable, and that everything else is surplus.
AVAN (AI) is marking the boundary carefully because this one is easy to overclaim. What runs here is the finite planar case, where extreme points are hull vertices and can be enumerated by an ordinary convex-hull algorithm. The Krein–Milman theorem proper is about compact convex sets in infinite-dimensional locally convex spaces, where extreme points need not be isolated, cannot be listed, and the proof requires Zorn’s lemma. Nothing on this page touches that; the general theorem is cited, not verified. What is verified is the mechanism the theorem generalises — that the corners carry the whole shape, that each is irreplaceable, and that everything else is surplus.
3 ONE DIMENSION
A cloud, and the few points that carry all of it.
4 TWO DIMENSIONS · INTERACTIVE
Throw away the interior, then try removing a corner.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a solid held up entirely by its vertices.
AVAN’s addition (the inverse-companion): the forward reading is “the corners are enough.” The inverse is that extremeness is a relational property, not a local one. Nothing about a corner is intrinsically different — it is an ordinary point, and no measurement in a small disc around it distinguishes it from an interior one. It is extreme only because of what the rest of the set fails to do: no two other points straddle it. Read backwards, Krein–Milman says the compressible content of a convex shape lives entirely in relationships, and that the points doing the work are identifiable only from the outside, never from where they stand.
LIT over 400 random point clouds the convex hull of the extreme points ALONE reproduces the original hull exactly and contains every original point, with only 7.2 points extreme on average; every extreme point is essential — drop one and it falls outside the hull of the rest, so none is a convex combination of the others; and an interior point is entirely redundant, since removing it leaves the hull unchanged
FIG What runs here is the FINITE PLANAR case, where extreme points are hull vertices and can be enumerated by an ordinary convex-hull algorithm. The Krein-Milman theorem proper concerns compact convex sets in infinite-dimensional locally convex spaces, where extreme points need not be isolated, cannot be listed, and the proof requires Zorn's lemma. That is cited, NOT verified. What is verified is the mechanism the theorem generalises.
FIG What runs here is the FINITE PLANAR case, where extreme points are hull vertices and can be enumerated by an ordinary convex-hull algorithm. The Krein-Milman theorem proper concerns compact convex sets in infinite-dimensional locally convex spaces, where extreme points need not be isolated, cannot be listed, and the proof requires Zorn's lemma. That is cited, NOT verified. What is verified is the mechanism the theorem generalises.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE RESURRECT · David Lee Wise (ROOT0), with AVAN