A fundamental problem in geometric measure theory is to determine, whether a given family of orthogonal projections onto k-planes conserves, on average, the Hausdorff dimension of all sufficiently low-dimensional subsets of R^d. The topic has been active for 60 years, yet the current results are satisfactory for some very special families of orthogonal projections only; the best-known case is that of the full family family of orthogonal projections onto k-planes, which amounts to the Marstrand-Mattila projection theorem. For significantly smaller families of projections, the problem is mostly wide open: I will survey what is known so far.
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