Abstract: Spherical integral transformations are indispensable tools in integral geometry and convex geometry. This paper gives a direct proof to the injectivity of p-cosine transformation forp=2k+1, k∈N. Using the injectivity properties of p-cosine transformation, an important result of uniqueness of star body is obtainted, i.e., a star body is uniquely determined by its p-centroid body forp=2k+1, k∈N

Key words:  star body, dual mixed volume, p-centroid body;p-cosine transform, spherical harmonic