In this article, I consider the right triangle as the simplex in the Euclidean plane, and extend this definition to higher dimensions. The n-dimensional simplex has one hypotenuse and (n−1)legs (catheti). The (n−1)leg...In this article, I consider the right triangle as the simplex in the Euclidean plane, and extend this definition to higher dimensions. The n-dimensional simplex has one hypotenuse and (n−1)legs (catheti). The (n−1)legs define an orthogonal path of edges in the solid with perpendicular adjacent edges along the path. The length of the hypotenuse and the volume of the solid can be calculated without the Cayley-Menger determinant, by direct extension of the corresponding right triangle formulas. I give a proof of the existence of these shapes, describe the distribution of right angles in them, give an algebraic proof of the Coxeter trisection of a right tetrahedron into three smaller right tetrahedra, and generalize this construction to n-dimensional spaces. Finally, I investigate the connection between the Coxeter partition and the Hadwiger conjecture on the partition of the simplex into orthoschemes, which I call Pythagorean simplexes.展开更多
In 1957,Hadwiger made a conjecture that every n-dimensional convex body can be covered by 2n translates of its interior.Up to now,this conjecture is still open for all n 3.In 1933,Borsuk made a conjecture that every n...In 1957,Hadwiger made a conjecture that every n-dimensional convex body can be covered by 2n translates of its interior.Up to now,this conjecture is still open for all n 3.In 1933,Borsuk made a conjecture that every n-dimensional bounded set can be divided into n + 1 subsets of smaller diameters.Up to now,this conjecture is open for 4 n 297.In this article we encode the two conjectures into continuous functions defined on the spaces of convex bodies,propose a four-step program to attack them,and obtain some partial results.展开更多
We estimate the kinematic measure of one convex domain moving to another under the group G of rigid motions in IR~n.We first estimate the kinematic formula for the total scalar curvature ∫Rdv of the n-2 dimensional i...We estimate the kinematic measure of one convex domain moving to another under the group G of rigid motions in IR~n.We first estimate the kinematic formula for the total scalar curvature ∫Rdv of the n-2 dimensional intersection submanifold D∩g D.Then we use Chern and Yen’s kinematic fundamental formula and our integral inequality to obtain a sufficient condition for one convex domain to contain another in IR~n(≥4).For n=4,we directly obtain another sufficient condition in IR~4.展开更多
A convex n-body C is said to be exposable to a set D of a few directions if there is a linear transformation L : En → En such that L(C) arid each body isometric to L(C) is illuminated by D. Denote by En the minimum i...A convex n-body C is said to be exposable to a set D of a few directions if there is a linear transformation L : En → En such that L(C) arid each body isometric to L(C) is illuminated by D. Denote by En the minimum integer such that each C is exposable to a (fixed) set D of cardinality En. An upper bound for En is established here.展开更多
In geometry, there are several challenging problems studying numbers associated to convex bodies. For example, the packing density problem, the kissing number problem, the covering density problem, the packing-coverin...In geometry, there are several challenging problems studying numbers associated to convex bodies. For example, the packing density problem, the kissing number problem, the covering density problem, the packing-covering constant problem, Hadwiger＇s covering conjecture and Borsuk＇s partition conjecture. They are flmdamental and fascinating problems about the same objects. However, up to now, both the methodology and the technique applied to them are essentially different. Therefore, a common foundation for them has been much expected. By treating problems of these types as functionals defined on the spaces of n-dimensional convex bodies, this paper tries to create such a foundation. In particular, supderivatives for these functionals will be studied.展开更多
文摘In this article, I consider the right triangle as the simplex in the Euclidean plane, and extend this definition to higher dimensions. The n-dimensional simplex has one hypotenuse and (n−1)legs (catheti). The (n−1)legs define an orthogonal path of edges in the solid with perpendicular adjacent edges along the path. The length of the hypotenuse and the volume of the solid can be calculated without the Cayley-Menger determinant, by direct extension of the corresponding right triangle formulas. I give a proof of the existence of these shapes, describe the distribution of right angles in them, give an algebraic proof of the Coxeter trisection of a right tetrahedron into three smaller right tetrahedra, and generalize this construction to n-dimensional spaces. Finally, I investigate the connection between the Coxeter partition and the Hadwiger conjecture on the partition of the simplex into orthoschemes, which I call Pythagorean simplexes.
基金supported by National Natural Science Foundation of China (Grant No.10225104)the Chang Jiang Scholars Program and LMAM at Peking University
文摘In 1957,Hadwiger made a conjecture that every n-dimensional convex body can be covered by 2n translates of its interior.Up to now,this conjecture is still open for all n 3.In 1933,Borsuk made a conjecture that every n-dimensional bounded set can be divided into n + 1 subsets of smaller diameters.Up to now,this conjecture is open for 4 n 297.In this article we encode the two conjectures into continuous functions defined on the spaces of convex bodies,propose a four-step program to attack them,and obtain some partial results.
文摘We estimate the kinematic measure of one convex domain moving to another under the group G of rigid motions in IR~n.We first estimate the kinematic formula for the total scalar curvature ∫Rdv of the n-2 dimensional intersection submanifold D∩g D.Then we use Chern and Yen’s kinematic fundamental formula and our integral inequality to obtain a sufficient condition for one convex domain to contain another in IR~n(≥4).For n=4,we directly obtain another sufficient condition in IR~4.
文摘A convex n-body C is said to be exposable to a set D of a few directions if there is a linear transformation L : En → En such that L(C) arid each body isometric to L(C) is illuminated by D. Denote by En the minimum integer such that each C is exposable to a (fixed) set D of cardinality En. An upper bound for En is established here.
基金Supported by 973 Programs(Grant Nos.2013CB834201 and 2011CB302401)the National Science Foundation of China(Grant No.11071003)the Chang Jiang Scholars Program of China
文摘In geometry, there are several challenging problems studying numbers associated to convex bodies. For example, the packing density problem, the kissing number problem, the covering density problem, the packing-covering constant problem, Hadwiger＇s covering conjecture and Borsuk＇s partition conjecture. They are flmdamental and fascinating problems about the same objects. However, up to now, both the methodology and the technique applied to them are essentially different. Therefore, a common foundation for them has been much expected. By treating problems of these types as functionals defined on the spaces of n-dimensional convex bodies, this paper tries to create such a foundation. In particular, supderivatives for these functionals will be studied.
