The radius of comparison of $C(X)$

Let X be a compact Hausdorff space. Then the radius of comparison rc ( C (X)) is related to the covering dimension dim (X) by rc ( C (X)) \geq [ dim (X) - 7 ] / 2. Except for the additive constant, this improves a result of Elliott and Niu, who proved that if X is metrizable then rc (C (X)) \geq [ dim_{$\mathbb{Q}$} (X) - 4 ] / 2. There are compact metric spaces X for which the estimate of Elliott and Niu gives no information, but for which rc ( C (X)) is infinite or has arbitrarily large finite values.