  We solve a long-standing open problem about the optimal codebook structure of
codes in $n$-dimensional Euclidean space that consist of $n+1$ codewords
subject to a codeword energy constraint, in terms of minimizing the average
decoding error probability. The conjecture states that optimal codebooks are
formed by the $n+1$ vertices of a regular simplex (the $n$-dimensional
generalization of a regular tetrahedron) inscribed in the unit sphere. A
self-contained proof of this conjecture is provided that hinges on symmetry
arguments and leverages a relaxation approach that consists in jointly
optimizing the codebook and the decision regions, rather than the codeword
locations alone.
