On the spherical slice transform

We study the spherical slice transform which assigns to a function f on the unit sphere Sn in n+1 the integrals of f over cross-sections of Sn by k-dimensional affine planes passing through the north pole (0,..., 0, 1). These transforms are known when k = n. We consider all 2 ≤ k ≤ n and obtain an explicit formula connecting with the classical (k - 1)-plane Radon-John transform Rk-1 on n. Using this connection, known facts for Rk-1, like inversion formulas, support theorems, representation on zonal functions, and some others, are reformulated for .