{"paper":{"title":"Shadows of convex bodies","license":"","headline":"","cross_cats":["math.FA"],"primary_cat":"math.MG","authors_text":"Keith Ball","submitted_at":"1989-10-26T14:59:00Z","abstract_excerpt":"It is proved that if $C$ is a convex body in ${\\Bbb R}^n$ then $C$ has an affine image $\\widetilde C$ (of non-zero volume) so that if $P$ is any 1-codimensional orthogonal projection,\n $$|P\\widetilde C| \\ge |\\widetilde C|^{n-1\\over n}.$$\n  It is also shown that there is a pathological body, $K$, all of whose orthogonal projections have volume about $\\sqrt{n}$ times as large as $|K|^{n-1\\over n}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9201204","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"}