Gini and Stolarsky means in geometric problems

1. Gini and Stolarsky means in geometric problems Alfred Witkowski University of Technology and Life Sciences, Bydgoszcz, Poland

2. What is n-frustum? Truncated cone with n-dimensional object as its base: Trapezoid is an 1-frustum El Castillo in Chichen Itza is a 2-frustum

3. Problem x How does the n-volume of selected horizontal sections (s) depend on n-volumes of its bases (x,y). Case n=1 was considered by Howard Eves in Means Appearing in Geometric Figures, Math. Magazine, 76, 4, (2001), 292-294 s y

4. x Cylinder with the same (n+1)-volume and height s s y Formula discovered (in case n=2) in 50 BC by Heron of Alexandria, that’s why we call them Heronian means.

5. x x Frusta of equal (n+1)-volumes s s y s y

6. x Equal heights x s s s ` y y

7. x x Similar frusta s s y s ! ! y

8. x x Equal lateral volume s s y s y

9. x Centroid(center of mass of solid frustum) s y

10. x x Center of mass of bases(or „inner” cones of equal (n+1)-volume) s s y y

11. x x Similar „inner” cones (or intersection of „diagonals”) s y y

12. x Lagrangean point s Point where gravitational attraction of x cancels that of y y

13. x (n+1)- volume of frustum equals sum of (n+1)-volumes of cylinders s y

14. x Cylinders of equal lateral volume s y

15. Order of means n>3 n=1 n=2 n=3 Lagrangean point Similar inner cones Similar frusta Equal heights Equal heights Cylinder of the same volume Frusta of equal lateral vol. Vol two cylinders=vol frustum Equal lateral vol of cylinders Cylinder of the same volume Vol two cylinders=vol frustum Centroid Frusta of equal lateral vol. Frusta of equal volume Equal lateral vol of cylinders Centers of masses

16. Homework