Jumps of the roundness function

Let $\rho$ be a parameter, let $\Lambda_\rho$ be the associated set, and let $R_\rho(\Lambda_\rho)$ denote its roundness. For each $k\in\mathbb{N}\setminus\{0,1\}$, consider the parameter values $\frac{1}{k}$ and $1-\frac{1}{k}$. Jumps of roundness function. The roundness function

$$\rho\mapsto R_\rho\left(\Lambda_\rho\right)$$

has jumps around $\frac{1}{k}$ and $1-\frac{1}{k}$ for any $k\in\mathbb{N}\setminus\{0,1\}$. This follows from the jumps of the function $\rho\mapsto d_\rho$ around these rational values together with the bounds established for the roundness.