科研之友
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摘要
For a convex body K in a Euclidean vector space X of dimension n (>= 2), we define two subarithmetic monotonic sequences {sigma(K,k)}(k >= 1) and {sigma(0)(K,k)}(k >= 1) of functions on the interior of K. The k-th members are "mean Minkowski measures in dimension k" which are pointwise dual: sigma(0)(K,k)(z) = sigma(z)(K),k(z), where z is an element of int K, and K-z is the dual (polar) of K with respect to z. They are measures of (anti-)symmetry of K in the following sense: @@@ 1 <= sigma(K,k)(z), sigma(0)(K,k)(z) <= k+1/2. @@@ The lower bound is attained if and only if K has a k-dimensional simplicial slice or simplicial projection. The upper bound is attained if and only if K is symmetric with respect to z. In 1953 Klee showed that the lower bound m*(K) > n - 1 on the Minkowski measure of K implies that there are n + 1 affine diameters meeting at a critical point z* is an element of K. In 1963 Grunbaum conjectured the existence of such a point in the interior of any convex body (without any conditions). While this conjecture remains open (and difficult), as a byproduct of our study of the dual mean Minkowski measures, we show that @@@ n/m(k)*+1 <= sigma(0)(K,n-1)(z*) @@@ always holds, and for sharp inequality Grunbaum's conjecture is valid.
- 出版日期2018-1
- 单位苏州科技大学
