  Context. The physical state of most of the baryonic matter in the local
universe is unknown, which is commonly referred to as the ``missing baryon
problem". It is theorized that at least half of these missing baryons are in a
warm-hot, low-density phase outside of the virialized dark-matter halos.
  Aims. We make an attempt to find the signature of this warm-hot intergalactic
medium (WHIM) phase in the filaments of the nearby Virgo cluster by using
optical and Sunyaev-Zeldovich effect data.
  Methods. Specifically, we use a filament-galaxy catalog created from the
HyperLeda database and an all-sky Compton-y map extracted from the Planck
satellite data for 2-dimensional cross-correlation analysis by applying
spherical harmonics transform. Significance test is based on the null-test
simulations which exploits advanced cut-sky analysis tools for a proper map
reconstruction. To place upper limits on the WHIM density in the Virgo
filaments, realistic baryonic density modelling within the cosmic filaments is
done based on state-of-the-art hydro-simulations, and it's done within the
signal-boosting routine.
  Results. The cross-correlation signal is found to be too dim compared to the
noise level in the Planck y-map. At 3$\sigma$ confidence level, the upper limit
on volume-average WHIM density turns out to be $\left\langle n_e \right\rangle
\lt 4\times10^{-4} cm^{-3}$, which is indeed consistent with the WHIM parameter
space as predicted from simulations.
