Polycyclic Aromatic Hydrocarbons (PAHs) are widely accepted as the carriers of the Aromatic Infrared Bands (AIBs), but an unambiguous identification of any specific interstellar PAH is still missing. For polar PAHs, pure rotational transitions can be used as spectral fingerprints for identification. Combining dedicated experiments, detailed simulations and observations, we explore d the mm wavelength domain to search for specific rotational transitions of corannulene (C20H10). We performed high-resolution spectroscopic measurements and a simulation of the emission spectrum of ultraviolet-excited C20H10 in the environment of the Red Rectangle (RR), calculating its synthetic rotational spectrum. Based on these results, we conducted a first observational campaign at the IRAM 30-m telescope towards this source to search for several high-J rotational transitions of C20H10. The laboratory detection of the J = 112 <- 111 transition of corannulene showed that no centrifugal splitting is present up to this line. Observations with the IRAM 30-m telescope towards the RR do not show any corannulene emission at any of the observed frequencies, down to a rms noise level of T-mb = 8 mK for the J =135 -> 134 transition at 137.615 GHz. Comparing the noise level with the synthetic spectrum, we are able to estimate an upper limit to the fraction of carbon locked in corannulene of about 1.0 x 10(-5) relative to the total abundance of carbon in PAHs. The sensitivity achieved in this work shows that radio spectroscopy can be a powerful tool to search for polar PAHs. We compare this upper limit with models for the PAH size distribution, emphasizing that small PAHs are much less abundant than predicted. We show that this cannot be explained by destruction but is more likely related to the chemistry of their formation in the environment of the RR.