Gerrymandering—the manipulation of electoral boundaries to maximize constituency wins—is often seen as a pathology of democratic systems. A commonly cited cure is to require that electoral constituencies have a ‘compact’ shape. But how much of a constraint does compactness in fact place on would-be gerrymanderers? We operationalize compactness as a convexity constraint and apply a theorem of Ka...

We study a generalization of relative submajorization that compares pairs positive operators on representation spaces some fixed group. A pair equivariantly relatively submajorizes another if there is an equivariant subnormalized channel takes the components first to satisfying similar positivity constraints as in definition submajorization. In context resource theory approach thermodynamics, t...