The Sharp Threshold for the Dyn-Farkhi Conjecture
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Statement
Dyn and Farkhi conjectured that the squared Hausdorff distance from a compact set to its convex hull is subadditive under Minkowski addition. It holds in dimensions one and two and fails from dimension three; the sharp threshold exponent for Hausdorff convexification is now determined.
Context
pins the sharp threshold; the conjecture itself was already known false above dimension two
A named conjecture in convex geometry with a documented dimension-by-dimension history, where the sharp exponent was the remaining question.
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The acknowledgement thanks ChatGPT 5.5 Pro for helping prove Theorem 1.4 and for generating the TikZ code for the figures, so the credit names a specific theorem rather than the paper as a whole.
pins the sharp threshold; the conjecture itself was already known false above dimension two
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