The aim of this paper is to prove an isoperimetric inequality relative to a d-dimensional, bounded, convex domain Ω intersected with balls with a uniform relative isoperimetric constant, independent of the size of the radius r>0 and the position y∈cl(Ω) of the center of the ball. For this, uniform Sobolev, Poincaré and Poincaré-Sobolev inequalities are deduced for classes of (not necessarily convex) domains that satisfy a uniform cone property. It is shown that the constants in all of these

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... ualities solely depend on the dimensions of the cone, space dimension d, the diameter of the domain and the integrability exponent p∈[1,d).