e_p^n空间中单位球迷向常数的极值问题

Extremum problems for isotropic constants of the unit balls in the spacen e_p^n

寻找凸体迷向常数的一致(与空间维数无关)上界是Banach空间局部理论中著名的公开问题.对于n p空间中单位球,作为1-无条件体的特例,其迷向常数一致上界的存在性是已知的.根据其已知迷向常数的解析表达式,利用动态优化的方法给出其精确的上、下界和处极值时相对应的几何体;再利用凸体迷向常数与超平面截片的等价性给出了一个对其中心截片极值问题的应用.

Finding the upper bound （independent on dimensional） of isotropic constants of convex bodies is a well known open problem in the local theory of Banach space. For unit balls in lp^n.space, that can be seen as the special case of 1-unconditional convex bodies, the existence of the upper bound of its isotropic constant is known. First, according to the known analytic expression of its isotropic constant, and by utilizing the method of dynamic programning, its precise upper and lower bound is given, and correspond geometric body of attaining extremum, again using equivalence of a convex body＇s central slicing and its isotropic constant, an application to lp^n. space unit ball＇s central slicing problem is given.