{"paper":{"title":"A Borsuk-Ulam theorem for digital images","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.GR"],"primary_cat":"math.GN","authors_text":"P. Christopher Staecker","submitted_at":"2015-06-22T00:04:04Z","abstract_excerpt":"The Borsuk-Ulam theorem states that a continuous function $f:S^n \\to \\R^n$ has a point $x\\in S^n$ with $f(x)=f(-x)$. We give an analogue of this theorem for digital images, which are modeled as discrete spaces of adjacent pixels equipped with $\\Z^n$-valued functions.\n  In particular, for a concrete two-dimensional rectangular digital image whose pixels all have an assigned \"brightness\" function, we prove that there must exist a pair of opposite boundary points whose brightnesses are approximately equal. This theorem applies generally to any integer-valued function on an abstract simple graph.\n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.06426","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"}