*Accepted Paper*

**Inserted:** 15 jan 2024

**Last Updated:** 5 jul 2024

**Journal:** J. Geom. Anal.

**Pages:** 37

**Year:** 2024

**Abstract:**

We prove a lower bound on the sharp PoincarĂ©-Sobolev embedding constants for general open sets, in terms of their inradius. We consider the following two situations: planar sets with given topology; open sets in any dimension, under the restriction that points are not removable sets. In the first case, we get an estimate which optimally depends on the topology of the sets, thus generalizing a result by Croke, Osserman and Taylor, originally devised for the first eigenvalue of the Dirichlet-Laplacian. We also consider some limit situations, like the sharp Moser-Trudinger constant and the Cheeger constant. As a byproduct of our discussion, we also obtain a Buser-type inequality for open subsets of the plane, with given topology. An interesting problem on the sharp constant for this inequality is presented.

**Keywords:**
Moser-Trudinger inequality, capacity, Torsional rigidity, Inradius, Buser's inequality, PoincarĂ©-Sobolev inequality, Cheeger's constant

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