  It is the Hilbert's Fourth Problem to characterize the
(not-necessarily-reversible) distance functions on a bounded convex domain in
R^n such that straight lines are shortest paths. Distance functions induced by
a Finsler metric are regarded as smooth ones. Finsler metrics with straight
geodesics said to be projective. It is known that the flag curvature of any
projective Finsler metric is a scalar function of tangent vectors (the flag
curvature must be a constant if it is Riemannian). Thus it is a natural problem
to study those of constant curvature.
  In this paper, we study the Hilbert Fourth Problem in the smooth case. We
first give a formula for x-analytic projective Finsler metrics F(x,y) of
constant curvature K=c using a power series with coefficients expressed in
terms of f(y):=F(0, y), h(y):=(1/2)F(x,y)^{-1}F_{x^k}(0, y)y^k and c. Then, for
any given pair {f(y), h(y)} and any constant c, we give an algebraic formula
for smooth projective Finsler metrics with constant curvature K=c and F(0,
y)=f(y) and F_{x^k}(0, y)y^k=2f(y)h(y). By these formulas, we obtain several
special projective Finsler metrics of constant curvature which can be used as
models in Finsler geometry.
