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在二维平面R~2内,众所熟知,有一个联系着平行四边形的周长和面积的定理,“周长一定的平行四边形中,以正方形的面积最大。”其证明极易由不等式 ab sinθ≤ab≤(a+b/2)~2 (△)给出(a、b均正数,θ∈(0,π,))。自然推想,在三维空间R~3中,是否有联系着平行六面体的表面积和体积的定理?即表面积一定的平行六面体中,以正立方体的体积最大?并由此进一步设想,在n维空间R~n中,关于平行2n面体,是否也存在一条类似的关于其表面积与体积的定理?结果本文在首先建立有关n阶行列式的两个不等式的基础上,非常方便地回答了上述问题。从而形成了同类问题的一个一般性定理;与此同时,由于文中所建立的两个不等式又恰为基本不等式(△)的推广,故必有其更广泛的理论意义。定理1:设
In the two-dimensional plane R~2, it is well known that there is a theorem that relates to the perimeter and area of parallelograms. “In the circumference of a certain parallelogram, the area of the square is the largest.” This proves easily by the inequality ab Sinθ≤ab≤(a+b/2)~2 (△) gives (a, b are positive, θ∈(0, π,)). Naturally, in the three-dimensional space R~3, is there a theorem of the surface area and volume of the parallelepiped? That is, in the parallelepiped with a certain surface area, the volume of the positive cube is the largest, and this is further conceived in the n-dimensional space R In ~n, is there a similar theorem about the surface area and volume of parallel 2n facets? The result This article is based on the establishment of two inequalities about the n-order determinant, and it is very convenient to answer the above questions. Thus, a general theorem of similar problems is formed; at the same time, because the two inequalities established in the paper are just the generalization of the basic inequality (△), it must have a broader theoretical significance. Theorem 1: Design