  We prove that quadratic forms in isotropic random vectors $X$ in
$\mathbb{R}^n$, possessing the convex concentration property with constant $K$,
satisfy the Hanson-Wright inequality with constant $CK$, where $C$ is an
absolute constant, thus eliminating the logarithmic (in the dimension) factors
in a recent estimate by Vu and Wang. We also show that the concentration
inequality for all Lipschitz functions implies a uniform version of the
Hanson-Wright inequality for suprema of quadratic forms (in the spirit of the
inequalities by Borell, Arcones-Gin\'e and Ledoux-Talagrand). Previous results
of this type relied on stronger isoperimetric properties of $X$ and in some
cases provided an upper bound on the deviations rather than a concentration
inequality.
  In the last part of the paper we show that the uniform version of the
Hanson-Wright inequality for Gaussian vectors can be used to recover a recent
concentration inequality for empirical estimators of the covariance operator of
$B$-valued Gaussian variables due to Koltchinskii and Lounici.
