Take any odd $C^\infty$-function $f$ on the sphere. Consider convex set $$R_\epsilon=\{\\,x\in\mathbb R^n\mid\langle x,u\rangle\le 1+\epsilon{\cdot}f(u)\ \ \text{for any}\ \ u\in\mathbb{S}^{n-1}\\,\}.$$ Clearly for all sufficiently small $\epsilon>0$, $R_\epsilon$ is a smooth body of constant width.

Take any odd $C^\infty$-function $f$ on the sphere. Consider the convex set $$R_\epsilon=\{\,x\in\mathbb R^n\mid\langle x,u\rangle\le 1+\epsilon{\cdot}f(u)\ \ \text{for any}\ \ u\in\mathbb{S}^{n-1}\,\}.$$ Note that for all sufficiently small $\epsilon>0$, $R_\epsilon$ is a smooth body of constant width.