In [A dozen de {F}inetti-style results in search of a theory, Ann. Inst. H. Poincar{e} Probab. Statist. 23(2)(1987), 397--423], Diaconis and Freedman studied low-dimensional projections of random vectors from the Euclidean unit sphere and the simplex in high dimensions, noting that the individual coordinates of these random vectors look like Gaussian and exponential random variables respectively. In subsequent works, Rachev and Ruschendorf and Naor and Romik unified these results by establishing a connection between $ell_p^N$ balls and a $p$-generalized Gaussian distribution. In this paper, we study similar questions in a significantly generalized and unifying setting, looking at low-dimensional projections of random vectors uniformly distributed on sets of the form [B_{phi,t}^N := Big{(s_1,ldots,s_N)inmathbb{R}^N : sum_{ i =1}^Nphi(s_i)leq t NBig},] where $phi:mathbb{R}to [0,infty]$ is a potential (including the case of Orlicz functions). Our method is different from both Rachev-Ruschendorf and Naor-Romik, based on a large deviation perspective in the form of quantitati
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