{"paper":{"title":"The Rainbow at the End of the Line --- A PPAD Formulation of the Colorful Carath\\'eodory Theorem with Applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"cs.CG","authors_text":"Fr\\'ed\\'eric Meunier, Pauline Sarrabezolles, Wolfgang Mulzer, Yannik Stein","submitted_at":"2016-08-05T15:45:28Z","abstract_excerpt":"Let $C_1,...,C_{d+1}$ be $d+1$ point sets in $\\mathbb{R}^d$, each containing the origin in its convex hull. A subset $C$ of $\\bigcup_{i=1}^{d+1} C_i$ is called a colorful choice (or rainbow) for $C_1, \\dots, C_{d+1}$, if it contains exactly one point from each set $C_i$. The colorful Carath\\'eodory theorem states that there always exists a colorful choice for $C_1,\\dots,C_{d+1}$ that has the origin in its convex hull. This theorem is very general and can be used to prove several other existence theorems in high-dimensional discrete geometry, such as the centerpoint theorem or Tverberg's theore"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.01921","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}