Derive closed-form formulas for the central front and general fibers

Derive closed-form formulas for Γρ and for the endpoints of the general vertical fiber Iρ(a, b) of the Carnot limit shape Rρ when 1 < ρ < 4.

Background

The paper gives an exact Wulff-duality formula for the origin fiber Iρ(0,0) below saturation and explicit formulas for the entire saturated shape when ρ ≥ 4. For general horizontal endpoints (a,b) below saturation, it proves that each nonempty vertical fiber Iρ(a,b) is a compact interval and provides quantitative upper and lower bounds, but not exact endpoint formulas.

The same remark identifies the absence of a closed-form expression for the central front Γρ in the sub-saturation regime as an unresolved problem. These formulas would provide more explicit geometric information about the nonconvex Carnot limit shape beyond the origin fiber and saturated cases.