CONICAL UNIQUENESS SETS FOR THE SPHERICAL RADON TRANSFORM M. L. AGRANOVSKY, V. V. VOLCHKOV L. A. ZALCMAN A Let K be a cone in n . Then K is a uniqueness set for the spherical Radon transform if and only if it is not contained in the zero set of any (nontrivial) homogeneous harmonic polynomial. A local version of this result is also proved. 1. Let f C( n ), n 2. For x n , r 0, the spherical Radon transform of f is defined by Rf (x, r) & S(x,r) fdσ, where S(x, r) denotes the (n1)-dimensional sphere of radius r centred at x, and σ is area measure on S(x, r). (The reader should be warned that there does not seem to be a standard terminology in this area. Some authors use spherical Radon transform to refer to the transform f (ω, t) defined below in Section 2, which we have called the spherical Radon transform on spheres.) A set U n is a uniqueness set for the spherical Radon transform if Rf (x, r) 0 for all r 0 and x U implies that f vanishes identically. In this paper, we are concerned with conical uniqueness sets, namely, uniqueness sets K which are also cones (so that if x K and λ 0, then λx K ). Answering a question of Palamodov, we prove the following. T 1. Let K be a cone in n . Then K is a uniqueness set for the spherical Radon transform if and only if it is not a subset of the zero set of any (nonzero) homogeneous harmonic polynomial or, equialently, of any harmonic function. The last equivalence is an easy consequence of the fact that any harmonic function can be expanded in a series of homogeneous harmonic polynomials (and that K is a cone). In fact, we shall prove the following local result, from which Theorem 1 follows immediately. Denote by B n the open unit ball in n . T 2. Let K B n , and suppose that λK K for any λ,0 λ 1. Let f C(B n ), and suppose that R f (x, r) 0 for all x K and all 0 r 1x. Then f 0 so long as K is not a subset of the zero set of any (nontriial ) homogeneous harmonic polynomial. For results on uniqueness sets having spherical symmetry, see [6]. Received 3 March 1998 ; revised 11 June 1998. 1991 Mathematics Subject Classification 44A12. Bull. London Math. Soc. 31 (1999) 231–236