In recent analysis we have defined and studied holomorphic functions in tubes in ℂ n which generalize the Hardy H p functions in tubes. In this paper we consider functions f ( z ) , z = x + i y , which are holomorphic in the tube T C = ℝ n + i C , where C is the finite union of open convex cones C j , j = 1 , … , m , and which satisfy the norm growth of our new functions. We prove a holomorphic extension theorem in which f ( z ) , z  ϵ  T C , is shown to be extendable to a function which is holomorphic in T 0 ( C ) = ℝ n + i 0 ( C ) , where 0 ( C ) is the convex hull of C , if the distributional boundary values in 𝒮 ′ of f ( z ) from each connected component T C j of T C are equal.