Carla Peri INTEGRAL GEOMETRY IN MINKOWSKI PLANE 1 ...

108rentiable and with positive finite curvature everywhere.For every v £ IR 2 let A E C be the unique point for which OA isparallel to v, as oriented vectors. The Minkowski length of v is defined by+ \v\(1) H *"-WT'where | • | denotes the Euclidean length.The points of IR 2 with the metric associated to the length (1) shall bereferred to as the Minkowski plane.If || — z>|| = ||?|| for any v G IR 2 (i.e. if C is a centrally symmetriccurve with center at the origin O) we shall refer to the plane as thesymmetric Minkowski plane.Let us denote the vectors from O to the indicatrix C by C. To themare associated vectors T(C) by [T(C), C] = 1, where [ , ] denotes thedeterminant. The vectors T(C), reported from O, describe an oval T calledisoperimetrix. The curve T is easily seen to be the polar reciprocal of C with7Trespect to the Euclidean unit circle, rotated through —r- .Let x{t) be a differentiable vector function, its Minkowski arclengtho x is the parameter defined up to an additive constant by(2)dxdo X= 1.We denote the Minkowski arclength of T by a and the area enclosedby T by II, then 0