{"paper":{"title":"On Euclidean $t$-designs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"B\\'ela Bajnok","submitted_at":"2015-12-09T18:29:39Z","abstract_excerpt":"A Euclidean $t$-design, as introduced by Neumaier and Seidel (1988), is a finite set ${\\cal X} \\subset \\mathbb{R}^n$ with a weight function $w: {\\cal X} \\rightarrow \\mathbb{R}^+$ for which $$\\sum_{r \\in R} W_r \\overline{f}_{S_{r}} = \\sum_{{\\bf x} \\in {\\cal X}} w({\\bf x}) f({\\bf x})$$ holds for every polynomial $f$ of total degree at most $t$; here $R$ is the set of norms of the points in ${\\cal X}$, $W_r$ is the total weight of all elements of ${\\cal X}$ with norm $r$, $S_r$ is the $n$-dimensional sphere of radius $r$ centered at the origin, and $\\overline{f}_{S_{r}}$ is the average of $f$ ove"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.02981","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"}