Zhenbing Zeng
ABSTRACT
This paper presents a new semi-mechanization method for proving the validity of an inequality conjecture about convex n-gon when n = 8 and gaining headway in proving it when n = 9. This conjecture is generally converted into a global optimization problem which is related to Heilbronn triangular problem. For solving it, the bottleneck is the complexity increasing very quickly with n. In the proposed algorithm, to reduce the dimension of freedom we first analyze the properties of the optimal configurations and try to obtain the strict polynomial inequality and equality conditions as many as possible. After the precondition, the mechanization method can be implemented to solve this nonlinear optimization problem, so we call the overall approach as semi-mechanization method. We hope our algorithm will be useful for proving the conjecture with larger value of n.
- L. Yang, Practical automated reasoning on inequalities: Generic programs for inequality proving and discovering, Proc. of The Third Asian Technology Conference in Mathematics, Springer-Verlag, 1998, 24--35.Google Scholar
- L. Yang and J. Zhang, A practical program of automated proving for a class of geometric inequalities, Automated Deduction in Geometry, Lecture Notes in Artificial Intelligence 2061, pp. 41--57, Springer-Verlag, 2001.Google Scholar
- Bottema O. et.al. Geometric Inequality. Groningen. 1969.Google Scholar
- A. W. M. Dress, L. Yang and Zhenbing Zeng, Heilbronn problem for six points in a planar convex body. Combinatorics and Graph Theory'95. vol 1 (Hefei), 97--118, World Sci. Publishing, River Edge, N.J., 1995.Google Scholar
- L. Yang, Jingzhong Zhang and Zhenbing Zeng, On Goldberg's conjecture: Computing the first several Heilbronn numbers. Universität Bielefeld, Preprint 91-074 (1991).Google Scholar
- L. Yang, Jingzhong Zhang and Zhenbing Zeng, On exact values of Heilbronn numbers for triangular regions. Universität Bielefeld, Preprint 91-098 (1991).Google Scholar
- L. Yang, Jingzhong Zhang and Zhenbing Zeng, A conjecture on the first several Heilbronn numbers and a computation, Chinese Ann. Math. Ser. A 13(1992), 503--515.Google Scholar
- M. Goldberg, Maximizing the smallest triangular made by N points in square, Math Magazine 45(1972), 135--144.Google ScholarCross Ref
- http://www.mmrc.iss.ac.cnGoogle Scholar
- http://www.acailab.com/dixon/download.htmGoogle Scholar
Index Terms
- Semi-mechanization method for a unsolved optimization problem in combinatorial geometry
Recommendations
Quantitative Combinatorial Geometry for Continuous Parameters
We prove variations of Carathéodory's, Helly's and Tverberg's theorems where the sets involved are measured according to continuous functions such as the volume or diameter. Among our results, we present continuous quantitative versions of Lovász's ...
Convex hulls of spheres and convex hulls of disjoint convex polytopes
Given a set @S of spheres in E^d, with d>=3 and d odd, having a constant number of m distinct radii @r"1,@r"2,...,@r"m, we show that the worst-case combinatorial complexity of the convex hull of @S is @Q(@__ __"1"=<"i"<>"j"=<"mn"in"j^@__ __^d^2^@__ __), ...
Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes
SoCG '11: Proceedings of the twenty-seventh annual symposium on Computational geometryGiven a set Σ of spheres in Ed, with d≥3 and d odd, having a fixed number of m distinct radii ρ1,ρ2,...,ρm, we show that the worst-case combinatorial complexity of the convex hull CHd(Σ) of Σ is Θ(Σ{1≤i≠j≤m}ninj⌊ d/2 ⌋), where ni is the number of ...
Comments