  Motivated by a wide-spread use of convex optimization techniques, convexity
properties of bit error rate of the maximum likelihood detector operating in
the AWGN channel are studied for arbitrary constellations and bit mappings,
which also includes coding under maximum-likelihood decoding. Under this
generic setting, the pairwise probability of error and bit error rate are shown
to be convex functions of the SNR and noise power in the high SNR/low noise
regime with explicitly-determined boundary. Any code, including
capacity-achieving ones, whose decision regions include the hardened noise
spheres (from the noise sphere hardening argument in the channel coding
theorem) satisfies this high SNR requirement and thus has convex error rates in
both SNR and noise power. We conjecture that all capacity-achieving codes have
convex error rates.
