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We show the existence of a sequence $(a_N)_{N\\ge 1}$ such that $a_N$ minimizes $a\\mapsto\\big \\|\\min_{1\\le i\\le N-1}|X-a_i|\\wedge |X-a|\\big\\|_{L^p}$ ($L^p$-mean quantization error at level $N$ induced by $(a_1,\\ldots,a_{N-1},a)$). We show that this sequence produces $L^p$-rate optimal $N$-tuples $a^{(N)}=(a_1,\\ldots,a_{_N})$ ($i.e.$ the $L^p$-mean quantization error at level $N$ induced by $a^{(N)}$ goes to $0$ at rate $N^{-\\frac 1d}$). 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We show the existence of a sequence $(a_N)_{N\\ge 1}$ such that $a_N$ minimizes $a\\mapsto\\big \\|\\min_{1\\le i\\le N-1}|X-a_i|\\wedge |X-a|\\big\\|_{L^p}$ ($L^p$-mean quantization error at level $N$ induced by $(a_1,\\ldots,a_{N-1},a)$). We show that this sequence produces $L^p$-rate optimal $N$-tuples $a^{(N)}=(a_1,\\ldots,a_{_N})$ ($i.e.$ the $L^p$-mean quantization error at level $N$ induced by $a^{(N)}$ goes to $0$ at rate $N^{-\\frac 1d}$). 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