Avkhadiev–Wirths conjecture on best Brezis–Marcus constants

We study Hardy-type inequalities with additional terms. The constant $\lambda(\Omega)$ multiplying the additional term depends on the geometry of the multidimensional domain $\Omega$ and the numerical parameters of the problem. This constant (functional) is commonly called the Brezis–Marcus constant. Avkhadiev and Wirths [1] put forward the conjecture that, over all $n$-dimensional domains with fixed inner radius $\delta_0$, the maximum best Brezis–Marcus constant is $\lambda(B_n)$, where $B_n $ is the $n$-ball of radius $\delta_0$. We improve the previously available lower estimates for $\lambda(B_n)$, for $n=2$ and $n= 4,\dots,10$, which takes us closer to this conjecture.
Bibliography: 18 titles.