Research interests

Convex Geometry

• The Covariogram Problem.
  To each regular compact set K in R^n one can associate a function g_K, called covariogram, or set covariance, of K. It can be defined as the function that to each x in R^n associates the volume of the intersection of K with its translate K+x.
  Does the knowledge of g_K determine K (Covariogram problem, G. Matheron 1986, R. Adler and R. Pyke, 1991 )? If g_K is radially symmetric, has K the same symmetry? Can one understand from g_K whether K is convex? What geometric information about K can one read in g_K?
  These questions are of interest also in Fourier analysis, Indeed, g_K coincides with the convolution of 1_K*1_-K.( 1_K denotes the characteristic function of K). Passing to Fourier transforms, knowing g_K is equivalent to knowing the modulus of the Fourier transform of 1_K. The Covariogram problem is thus an example of a Phase Retrieval problem, and is relevant also in X-ray crystallography, in the determination of certain materials, the "quasi-crystals", from their diffraction images.


• Slides on the Covariogram problem (conference talk given in Cortona in 2011, updated in 2020)


• Symmetrizations of sets and convergence of iterates of symmetrizations.
  A symmetrization, with respect to a given subspace H, is simply an operation that transform a set, for instance compact or convex, in an H-symmetric set in the same class. In many cases of interest the symmetrization maintain certain functionals and monotonically changes some others. The most famous examples are the Steiner symmetrization, which maintain the volume and, in general, reduces surface area, and the Minkowski symmetrization, which does not change the mean width and, in general, increases volume and surface area.
  These operations are at the basis of the rearrangements of functions, where each level set is substituted by its symmetrization.
  A key ingredient in this is also the behaviour of the symmetrizations under iterations, and the possibility of finding a sequence of subspaces so that the corresponding sequence of iterates of a set converges, possibly to a ball. Steiner used these ingredients to give a proof of the isoperimetric inequality in the class of convex bodies.


• Slides on Symmetrizations (conference talk given in Pisa in 2019)

Partial Differential Equations

• Scalar curvature problem.
  Problems of existence, non-existence and symmetry of solutions of some elliptic semilinear PDE which arise in differential geometry. An example is the scalar curvature problem (or Kazdan Warner problem), where one tries to understand for which functions K(x) there exists a metric on Sⁿ conformal to the standard one and whose scalar curvature is K(x). This problem reduces to the problem of the existence of a solution to an elliptic equation on Rⁿ with a nonlinearity related to the so-called ''critical exponent''.


• ''Bianchi-Egnell'' stability estimate for the Sobolev inequality.
  A positive answer to a question asked by H. Brezis e E. Lieb regarding the existence of a "remainder term" in the Sobolev inequality associated to the immersion of D^{1,2}(R^n) in L^{2n/(n-2)}(R^n): the difference between the L^2 norm of the gradient of a function u and its L^{2n/(n-2)} norm can be bounded from below by a positive constant times the distance of u from the manifold of the functions for which the Sobolev inequality holds with equality.