Crofton-type integral formulas for star domain
Let C be a star-shaped curve in the Euclidean plane, i.e. let C be simply closed and bounding a domain D, star-shaped with respect to a point 0. First a special representation of differentiable star-shaped curves is introduced. Then the following extremal problem for star-shaped curves is treated: Find a straight line through 0 dividing D into extremal parts with respect to the area. Finally some simple Crofton-type integral formulas concerning straight lines through 0 and expressing e.g. the length of C and the area of D are derived.
scientific article; zbMATH DE number 4086476