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The generalized mutual information (GMI) of bit-interleaved coded modulation (BICM) systems, sometimes called the BICM capacity, is investigated at low signal-to-noise ratio (SNR). The combinations of input alphabet, input distribution, and binary labeling that achieve the Shannon limit - 1.59 dB are completely characterized. The main conclusion is that a BICM system with probabilistic shaping achieves the Shannon limit at low SNR if and only if it can be represented as a zero-mean linear projection of a hypercube. Hence, probabilistic shaping offers no extra degrees of freedom to optimize the low-SNR BICM-GMI, in addition to what is provided by geometrical shaping. The analytical conclusions are confirmed by numerical results, which also show that for a fixed input alphabet, probabilistic shaping can improve the BICM-GMI in the low and medium SNR range.