By B. Guenin

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Extra resources for Combinatorial Optimization [Lecture notes]

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This means that in each projective Finsler geometry one has a version of the Crofton formula. The following exercise describes a result of Hamel, a student of Hilbert, obtained in 1901, shortly after Hilbert’s ICM talk. 22. A Lagrangian L(x1 , x2 , v1 , v2 ) defines a projective Finsler metric if and only if ∂2L ∂2L = . 23. A “magnetic” version of Hilbert’s fourth problem is considered in [115], where Finsler metrics in the plane are described such that their geodesics are circles of a fixed radius.

1 Hint: Assume, more generally, that one randomly drops a curve on the ruled paper. The average number of intersections with a line depends only on the length of the curve and equals 2 for a circle of diameter 1 whose perimeter length is π. The Crofton formula has numerous applications; see [89]. We will discuss four. 2), and let l and L be their lengths. We claim that L ≥ l. Indeed, a line intersects a convex curve at two points, and every line that intersects the inner curve intersects the outer one as well.

In the case at hand, we start with the cotangent bundle M = T ∗ Rn and its canonical symplectic structure ω. Let (x, y) be coordinates in T ∗ Rn (instead of (q, p) which will be used as coordinates in the space of lines) so that ω = dx ∧ dy. The hypersurface S consists of unit (co)vectors |y|2 = 1. Hence the 1-form ydy vanishes on S. Given a unit tangent vector (x, y), the respective rectilinear motion is described by the vector field y∂x. Let ξ be an arbitrary tangent test vector to S; then (dx ∧ dy)(y∂x, ξ) = (ydy)(ξ) = 0 since ydy = 0 on S.

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Combinatorial Optimization [Lecture notes] by B. Guenin

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