Uniqueness of the measurement function in Croftons formula Abstract   Croftonu0027s intersection formula states that the    (  n  xe2x88x92  j  )   th intrinsic volume of a compact convex set in      R    n      can be obtained as an invariant integral of the    (  k  xe2x88x92  j  )   th intrinsic volume of sections with kplanes. This paper discusses the question if the    (  k  xe2x88x92  j  )   th intrinsic volume can be replaced by other functionals, that is, if the measurement function in Croftonu0027s formula is unique.  The answer is negative: we show that the sums of the    (  k  xe2x88x92  j  )   th intrinsic volume and certain translation invariant continuous valuations of homogeneity degree k yield counterexamples. If the measurement function is local, these turn out to be the only examples when    k    1    or when    k    2    and we restrict considerations to even measurement functions. Additional examples of local functionals can be constructed when    k  xe2x89xa5  2   .

Symmetry from sectional integrals for convex domains Abstract   Let xcexa9 be a bounded convex domain in      R    n      (   n  xe2x89xa5  2   ). In this work, we prove that if there exists an integrable function f such that itu0027s Radon transform over    (  n  xe2x88x92  1  )   dimensional hyperplanes intersecting the domain xcexa9 is a function G depending on the distance to the nearest parallel supporting hyperplane to xcexa9, then xcexa9 is a ball and f is radial depending on certain assumptions on G. As a consequence we show that constants are not in range of Radon transform of integrable functions in dimension    n  xe2x89xa5  3   .