  It is known that an $n$-dimensional convex body which is typical in the sense
of Baire category, shows a simple, but highly non-intuitive curvature
behaviour: at almost all of its boundary points, in the sense of measure, all
curvatures are zero, but there is also a dense and uncountable set of boundary
points at which all curvatures are infinite. The purpose of this paper is to
find a counterpart to this phenomenon for typical convex bodies of given
constant width. Such bodies cannot have zero curvatures. A main result says
that for a typical $n$-dimensional convex body of constant width $1$ (without
loss of generality), at almost all boundary points, in the sense of measure,
all curvatures are equal to $1$. (In contrast, note that a ball of width $1$
has radius $1/2$, hence all its curvatures are equal to $2$.) Since the
property of constant width is linear with respect to Minkowski addition, the
proof requires recourse to a linear curvature notion, which is provided by the
tangential radii of curvature.
