The mid-facet of a simplex in n-dimensional Euclidean space which was introduced quite recently is an important geometric element. An analytic expression for the mid-facet area of a simplex is firstly given. In order ...The mid-facet of a simplex in n-dimensional Euclidean space which was introduced quite recently is an important geometric element. An analytic expression for the mid-facet area of a simplex is firstly given. In order to obtain the expression,the exterior differential method was presented. Furthermore, the properties of the mid-facets of a simplex analogous to median lines of a triangle (such as for all mid-facets of a simplex,there exists another simplex such that its edge-lengths equal to these mid-facets area respectively, and all of the mid-facets of a simplex have a common point) were proved. Finally, by applying the analytic expression, a number of inequalities which combine edge-lengths, circumradius, median line, bisection area and facet area with the mid-facet area for a simplex were established.展开更多
设K是Rn中体积为1,质心在原点的凸体,LK是它的迷向常数,寻找LK的上确界,是Banach空间局部理论(现代几何分析)中著名的未解决问题.目前最好的上界估计是LK<cn1/4logn,它是由Bourgain证明的.最近,何斌吾、冷岗松又证明了当r1Bn2 K r2Bn...设K是Rn中体积为1,质心在原点的凸体,LK是它的迷向常数,寻找LK的上确界,是Banach空间局部理论(现代几何分析)中著名的未解决问题.目前最好的上界估计是LK<cn1/4logn,它是由Bourgain证明的.最近,何斌吾、冷岗松又证明了当r1Bn2 K r2Bn2(r1≥1/2,r2≤n/2)时,LK≤1/(2 3),并猜测在对称几何体中以超立方体的迷向常数为最大,在非对称几何体中以单形的迷向常数为最大.给出了在三维空间中全部正多面体的迷向常数的数值,从而说明这一猜测对三维空间中的正多面体是正确的.展开更多
文摘The mid-facet of a simplex in n-dimensional Euclidean space which was introduced quite recently is an important geometric element. An analytic expression for the mid-facet area of a simplex is firstly given. In order to obtain the expression,the exterior differential method was presented. Furthermore, the properties of the mid-facets of a simplex analogous to median lines of a triangle (such as for all mid-facets of a simplex,there exists another simplex such that its edge-lengths equal to these mid-facets area respectively, and all of the mid-facets of a simplex have a common point) were proved. Finally, by applying the analytic expression, a number of inequalities which combine edge-lengths, circumradius, median line, bisection area and facet area with the mid-facet area for a simplex were established.
文摘设K是Rn中体积为1,质心在原点的凸体,LK是它的迷向常数,寻找LK的上确界,是Banach空间局部理论(现代几何分析)中著名的未解决问题.目前最好的上界估计是LK<cn1/4logn,它是由Bourgain证明的.最近,何斌吾、冷岗松又证明了当r1Bn2 K r2Bn2(r1≥1/2,r2≤n/2)时,LK≤1/(2 3),并猜测在对称几何体中以超立方体的迷向常数为最大,在非对称几何体中以单形的迷向常数为最大.给出了在三维空间中全部正多面体的迷向常数的数值,从而说明这一猜测对三维空间中的正多面体是正确的.
