Hilbert’s Fourth Problem by Aleksei V. Pogorelov

By Aleksei V. Pogorelov

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The metric specified by this line element is a a-metric. In fact, the expression J y ( g ) \% dx I d \ Cl) represents the value of the function a, defined by the formula a — JY dp d \, 28 HILBERT’S FOURTH PROBLEM on the set of lines intersecting the element dx (see Note (10) for the calculation). Let Fi (x,x ) = Jy|i*| d t To prove the Theorem, we need only show that F = cF x, where c is a constant. To this end, we take a small isosceles triangle uvw , with base uv o f euclidean length 6 on the line g, and small base angles, equal to a.

The distance between the points x and x + a • % appears on the left-hand side of this equation, while the length of the segment joining these points, measured in the metric p ', appears on the right-hand side. Let 7 be a smooth curve. Then the length of у is the same in both the metric p # and the metric p '. To see this, we note that, by definition, the length of the curve у in the metric pa is given by L(y) = sup pa Xk). AVERAGING DESARGUESIAN METRICS 37 It follows that there exists a sequence of partitions of the curve 7 , by points XV; such that L{i ) = Hm 2 ]p « {Xk-i, л " ).

The function F is positive homogeneous of degree I and convex. In fact, F (Xx' + цх") = sup у (Xx' + цх") ует < X sup у х ' + р sup у х " = X F (Xf) + fiF (х") уеТ уsT for arbitrary X, p ^ 0. Conversely, if a function F(x) is positive homogeneous of degree I and convex, then it is the support function 55 FUNK’S PROBLEM of some convex body. This convex body is the intersection of the half-spaces Ey defined by the inequalities x y < F (y). Let у (I) be a nonnegative even function of the unit vector | , defined on the sphere

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