On the reductibility of the general existence problem for closed convex surfaces to Miranda's equation

Abstract

It is proved that the solution of the general existence problem for closed convex surfaces with prescribed local propertiesf(R1R2,R1+R2,n)=ϕ(n) can be obtained as the solution of Miranda's equationR1R2+Φ(f)+cn=ψ(φ(n),ϕ(n)) with right-hand side depending on the unknown surface under the hypothesis that the latter satisfies the “closure condition”\(\int\limits_\Omega {n\phi (n)d} \omega = 0\), where μ is the unit sphere anddω is its element of area.