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Let    n  ⩾  C    for a large universal constant    C  >  0    and let   B   be a convex body in    R  n    such that for any    (   x  1   ,   x  2   ,  …  ,   x  n   )  ∈  B   , any choice of signsɛ  1   ,   ɛ  2   ,  …  ,   ɛ  n   ∈   {  −  1  ,  1  }and for any permutation   σ   on   n   elements, we have    (   ɛ  1    x   σ  (  1  )    ,   ɛ  2    x   σ  (  2  )    ,  …  ,   ɛ  n    x   σ  (  n  )    )  ∈  B   . We show that if   B   is not a cube, then   B   can be illuminated by strictly less than    2  n    sources of light. This confirms the Hadwiger–Gohberg–Markus illumination conjecture for unit balls of   1  ‐symmetric norms in    R  n    for all sufficiently large   n  .

ILLUMINATION OF CONVEX BODIES WITH MANY SYMMETRIES