Let Σ be a closed surface (i.e. a 2-dimensional Riemannian manifold) satisfying the following condition: the first eigenvalue of the elliptic operator -Δ+βK is nonnegative, where K is the Gauss curvature and β is a positive constant . This condition was mainly motivated by the studies of positive scalar curvature in dimension three, and soon showed its own interest. This condition can be understood as a global and weak condition on the positivity of curvature. Therefore, it is natural to ask whether the surface Σ in question shows any geometric characteristic of positive curvature. We try to answer this problem in the perspective of isoperimetric inequalities. The main results are as follows. When β>1/2, the Cheeger constant of Σ is bounded from below in terms of the diameter. When β>1/4, the homogeneous isoperimetric ratio of Σ is bounded from below in terms of the diameter and total area of Σ. We also discuss the optimality of the bounds on β that appeared in the results.