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For each    n  ⩾  2    we construct a measurable subset of the unit ball in    R  n    that does not contain pairs of points at distance 1 and whose volume is greater than(  1  /  2  )   n    times the volume of the unit ball. This disproves a conjecture of Larman and Rogers from 1972.

A COUNTEREXAMPLE TO A CONJECTURE OF LARMAN AND ROGERS ON SETS AVOIDING DISTANCE 1