This is hardly a direct answer to your question, but a new paper—at least tangentially relevant—by HaiLin Jin and Qi Guo addresses the question of how assymetric can a constant-width body be. In "Asymmetry of Convex Bodies of Constant Width" (Discrete & Computational Geometry Vol. 47, No. 2, Mar. 2012, 415-423), they establish tight bounds on the "Minkowski measure of asymmetry for convex bodies." In particular, they extend the known result that Reuleaux triangles are most assymetric in $\mathbb{R}^2$ to showing that Meissner's tetrahedron is most assymetric in $\mathbb{R}^3$. An image of Meissner's tetrahedron appeared in the earlier MO question, "Are there smooth bodies of constant width?width?"
This is hardly a direct answer to your question, but a new paper—at least tangentially relevant—by HaiLin Jin and Qi Guo addresses the question of how assymetric can a constant-width body be. In "Asymmetry of Convex Bodies of Constant Width" (Discrete & Computational Geometry Vol. 47, No. 2, Mar. 2012, 415-423), they establish tight bounds on the "Minkowski measure of asymmetry for convex bodies." In particular, they extend the known result that Reuleaux triangles are most assymetric in $\mathbb{R}^2$ to showing that Meissner's tetrahedron is most assymetric in $\mathbb{R}^3$. An image of Meissner's tetrahedron appeared in the earlier MO question, "Are there smooth bodies of constant width?"