Isodiametric-type upper bounds for Steklov eigenvalues

In this paper, we establish isodiametric-type upper bounds for Steklov eigenvalues of bounded Lipschitz domains $Ω\subset \mathbb{R}^n$, $n\geq 2$, valid for every $k\in \mathbb{N}$: \[ D(Ω)σ_k(Ω)\leq C(n)k, \] \[ D(Ω)σ_k(Ω)\leq C(n)k^2\left(\frac{|Ω|^{\frac{1}{n}}}{D(Ω)}\right)^{\frac{n}{n-1}}, \]