关键词:
chord power integral
containment function
distribution function
moments of random chord
probability density
摘要:
Using the chord power integral and its inequalities of a convex body, we establish inequalities about moments for μ-random chord length, ν-random chord length, and A-random chord length in ℝn. Based on the relationship between the chord power integral and containment function of a convex body, we obtain a new expression for moments of three kinds of random chord length mentioned above. By utilizing the properties of the distribution function and probability density function of μ-random chord length, we get the calculation formulas for the distribution function and probability density function of ν-random chord length, and the distribution function and probability density function of A-random chord length, respectively. Further, we establish the relationships among three kinds of distribution functions. On this basis, taking a rhombus, regular pentagon, and regular hexagon as examples in R2, we give the expressions of their 1-order moment for three kinds of random chord length and the distribution function of ν-random chord length. © 2025 Chinese Academy of Sciences. All rights reserved.