Asymmetry of convex bodies of constant width (Q664358)

The symmetry of convex bodies of constant width is discussed. It is shown that the Minkowski measure of asymmetry \(as_\infty (K)\) of an arbitrary convex body \(K\subset \mathbb{R}^n\) of constant width satisfies \(1\leq as_\infty(K)\leq \frac{n+\sqrt{2n(n+1)}}{n+2}\). The equality holds on the left-hand side if and only if \(K\) is an Euclidean ball. The upper bounds are attainable too: in particular, if \(n=3\), then the equality holds on the right-hand side if \(K\) is a Meissner body.