{"id":"palomar/PALOMAR-2026-08-29-000012","url":"https://github.com/sneed-and-feed/lean-theorems-1/tree/737f479090feab036d6f83b069e23a9e523daf33","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-08-29-000012-v1.json","repo":"https://github.com/sneed-and-feed/lean-theorems-1","tags":["math.MG","math.CO","msc:52A35","msc:52A20","msc:05D15","convex-geometry","lean-formalization","caratheodory-theorem","discrete-geometry"],"score":0,"title":"sneed-and-feed/lean-theorems-1","trust":"high","domain":"math.MG","source":"palomar","authors":["Sneed & Feed Formalization Team"],"reviews":0,"subject":"math.MG","summary":"A complete, machine-checked Lean 4 formalization of Bárány's Colorful Carathéodory Theorem (1982), Bárány's First Selection Lemma (1982), the 1D Centerpoint Theorem, and geometric deductions from first principles. Proves that for any d + 1 sets of points S_0, ..., S_d in R^d whose convex hulls all contain a common target point p, there exists a colorful transversal f (selecting f(i) in S_i for each color class i) whose convex hull conv({f(0), ..., f(d)}) contains p. Formulated over arbitrary dimension d, with specializations to the origin and low dimensions (d = 0, 1, 2), a formal deduction…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-08-29","reactions":{},"formalized":true,"formalizes":[{"title":"A generalization of Carathéodory's theorem","identifier":"","relationship":"formalizes"},{"title":"Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen","identifier":"","relationship":"adapts"},{"title":"A Theorem on General Measure","identifier":"","relationship":"adapts"}]}