Geometric properties of Euclidean domains supporting trace inequalities

We investigate the geometric behavior of $\tau(E)$ for bounded finite-perimeter sets $E \subset \mathbb R^n$, where $\tau(E)$ is the trace constant introduced by Figalli--Maggi--Pratelli [Invent. Math. 2010]. This quantity is a key ingredient in proving a quantitative isoperimetric inequality with the optimal exponent. We first show that for every $\epsilon>0$ one can find a bounded open set $\Omega \subset \mathbb R^n$ that is very close to the unit ball $\mathbb B^n$ in the sense that $$\tau(\mathbb B^n)>\tau(\Omega)>\tau(\mathbb B^n)-\epsilon \quad \text{and} \quad P(\Omega \Delta \mathbb B^n)\le C(n)\epsilon,$$ while at the same time the complement of $\Omega$ has infinitely many connected components. Thus, $\tau(\Omega)$ can be made arbitrarily close to $\tau(\mathbb B^n)$ even when $\Omega$ has highly intricate geometry. We then establish, under a mild additional hypothesis, the equivalence between a condition formulated in terms of $\tau$ and two classical criteria from the literature for open sets that admit trace inequalities. As a consequence, we obtain the John-type characterization of domains that support a trace inequality, assuming the ball separation property.