We investigate the convexity of the radial sum of two convex bodies containing the origin. Generally, the radial sum of two convex bodies containing the origin is not convex. We show that the radial sum of a star body (with respect to the origin) and any large centered ball is convex, which produces a pair of convex bodies containing the origin whose radial sum is convex.

We also investigate the convexity of the intersection body of a convex body containing the origin. Generally, the intersection body of a convex body containing the origin is not convex. Busemann’s theorem states that the intersection body of any centered convex body is convex. We are interested in how to construct convex intersection bodies from convex bodies without any symmetry (especially, central symmetry). We show that the intersection body of the radial sum of a star body (with respect to the origin) and any large centered ball is convex, which produces a convex body with no symmetries whose intersection body is convex.

我们研究包含原点的两个凸体的径向和的凸性。一般来说，包含原点的两个凸体的径向和不是凸的。我们证明，星体（相对于原点）和任何大的中心球的径向和是凸的，这会产生一对包含原点的凸体，其径向和是凸的。

我们还研究了包含原点的凸体的交集体的凸性。一般来说，包含原点的凸体的交集不是凸体。布斯曼定理指出，任何中心凸体的交集都是凸体。我们感兴趣的是如何从没有任何对称性（特别是中心对称）的凸体构造凸交体。我们证明，星体（相对于原点）和任何大的中心球的径向和的交体是凸的，这产生了一个没有对称性的凸体，其交体是凸的。