ILLUMINATION of CONVEX BODIES with MANY SYMMETRIES

Let for a large universal constant 0> and let be a convex body in such that for any , any choice of signs and for any permutation on elements, we have . We show that if is not a cube, then can be illuminated by strictly less than sources of light. This confirms the Hadwiger-Gohberg-Markus illumination conjecture for unit balls of -symmetric norms in for all sufficiently large .