凸体的L_p-径向线性组合

L_p-radial Linear Combinations of Convex Bodies

结合径向线性组合的定义给出了凸体的Lp-径向线性组合的定义,并探讨出了该组合下凸体的对偶混合体积之间的大小关系,重点研究了凸体的Lp-径向线性组合的平均宽度的相关性质,同时还给出了凸体的Lp-径向线性组合和它的极体的平均宽度的下界。在充分研究凸体的Lp-径向线性组合性质的基础上,得到凸体的Lp-径向线性组合和它的Firey线性组合之间的一些关系式。最后,在这些定理的支撑下,得到了关于凸体的Firey线性组合的平均宽度的下界的一般性结论,若K,L∈K0n实数p≥1,λ,μ≥0,λ+μ=1,则M((λK*+pμL*)+(λK+pμL))≥4,等号成立当且仅当K与L都为单位球时。

There are so many researches and papers for radial linear combinations of convex bodies, and many beautiful results of its properties are gotten. Based on these theories, the Lp-radial linear combinations of convex bodies are introduced. In this paper we deduce the size of these dual mixed volumes, and some properties for their mean width are studied. Meanwhile, the lower bound of the mean width of Lp-radial linear combinations of convex bodies and their polar bodies are found. After many properties for Lp-radial linear combinations of convex bodies are studied enough, we have found the connection between the Lp-radial linear combinations and Firey linear combinations of convex bodies. Finally, supported by these theories, we got the lower bound of the mean width of Firey linear combinations of convex bodies which are a more general form： For K,L∈K0^n, real number p≥1,λ,μ≥0,λ＋μ=1,then M（λK^＊＋pμL^＊）＋（λK＋pμL））≥4, with equality if and only if both K and L are unit balls.