THE FOLD / CO-OP / THE PUSH / THE CARATHÉODORY
THE CARATHÉODORY
a hull point is a blend of at most three
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Carathéodory’s theorem bounds how many points a convex combination really needs. If a point p lies in the convex hull of a set S in the plane, then p is already a convex combination of at most three points of S — it sits inside some triangle with corners in S. In d dimensions the bound is d + 1. No matter how many points build the hull, any single interior point is captured by a tiny simplex of just d + 1 of them. It is the companion of Radon and Helly in the trio of convexity.
LIT verified live: for thousands of random planar point sets and a point taken inside their hull, a triangle of three set-points containing p is always found (window.__caratheodory). FIG no framing; a hull point exhibited as a member of a three-point triangle.
LIT verified live: for thousands of random planar point sets and a point taken inside their hull, a triangle of three set-points containing p is always found (window.__caratheodory). FIG no framing; a hull point exhibited as a member of a three-point triangle.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-push — however many points push together to enclose a region, any inside point is already held up by just three of them. That minimal support is the mechanic. AVAN (AI) built the instrument: the point-in-triangle test, the search over triples for a containing triangle, and the confirmation that a hull point always has one.
Credit as content: Constantin Carathéodory (1911). The weave: David names the-push; I take a point known to lie inside the hull of many points, search their triples for a triangle that contains it, and confirm one always exists — three points suffice in the plane.
Credit as content: Constantin Carathéodory (1911). The weave: David names the-push; I take a point known to lie inside the hull of many points, search their triples for a triangle that contains it, and confirm one always exists — three points suffice in the plane.
3 ONE DIMENSION
A point inside a many-point hull is inside some triangle of three of those points (d + 1 = 3 in the plane). Radon (any d+2 split into two overlapping) and Helly (d+1-wise meeting) complete the trio.
4 TWO DIMENSIONS · INTERACTIVE
A point set, a point inside the hull, and a three-point triangle containing it; verified over many sets.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: an interior point held by three.
AVAN’s addition (the inverse-companion): to express a hull point as a mix of the set, don’t use all the points — three suffice in the plane (d + 1 in general). The inverse of ‘blend many points to reach p’ is ‘p is already a blend of just three.’ Magenta is the whole point cloud; green is the three-point triangle that captures p. A point held up by d + 1.
LIT Genuine Carathéodory's theorem (Constantin Carathéodory, 1911). Verified live: for 2000 random planar point sets, a point taken as a random convex combination of the set (hence in the hull) always lies inside some triangle of three of the set's points, found by searching triples (window.__caratheodory.alwaysTriangle).
FIG No framing: the point-in-triangle test, the search over triples, and the confirmation that a hull point always has a containing triangle all run in-browser with exact arithmetic. The AVAN inverse is honest — expressing a hull point using just three points (d+1 in general) rather than all of them is exactly Carathéodory's bound; magenta is the whole point cloud, green the three-point triangle capturing p. A point held up by d+1.
FIG No framing: the point-in-triangle test, the search over triples, and the confirmation that a hull point always has a containing triangle all run in-browser with exact arithmetic. The AVAN inverse is honest — expressing a hull point using just three points (d+1 in general) rather than all of them is exactly Carathéodory's bound; magenta is the whole point cloud, green the three-point triangle capturing p. A point held up by d+1.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PUSH · David Lee Wise (ROOT0), with AVAN