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OP-7574Status unassessedCombinatorics

Approximate Caratheodory theorem

Additive combinatorics and its applications · Historical workshop question

MSC 05 · Combinatorics

Editorial mapping of AIMPL source category: Combinatorics. http://aimpl.org/addcombapp/ Not an author-supplied MSC code.

Historical list · status unassessedAs of 2026-09-08Source & evidence ↓
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Problem statement

Q. First, assuming some assumptions on the point set AA, find a better dependence (sub-linear) on pp.

Q. Characterize configurations for which the bound O(pε−2)O(p\varepsilon^{-2}) is tight. status

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Background

Question context

There is the following important theorem: Let A⊂RdA\subset \mathbb{R}^d be a finite set of points. Also assume that AA is contained in the LpL^p unit Hamming ball for p≥2p\geq 2. Then any arbitrary point aa in the convex hull of AA can be approximated by a convex combination of only O(pε−2)O(p\varepsilon^{-2}) many points of AA.

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References
  1. [1]Additive combinatorics and its applications link

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