Note on illuminating constant width bodies

Recently, Arman, Bondarenko, and Prymak constructed a constant width body in ${\mathbb{R}}^n$ whose illumination number is exponential in $n$. In this note, we improve their bound by generalizing the construction. In particular, we construct a constant width body in ${\mathbb{R}}^n$ whose illumination number is at least $(\tau +o(1){)}^n$, where $\tau \approx 1.047$.