  In this paper, we show that if $L_p$ Gaussian surface area measure is
proportional to the spherical Lebesgue measure, then the corresponding convex
body has to be a centered disk when $p\in[0,1)$. Moreover, we investigate $C^0$
estimate of the corresponding convex bodies when the density function of their
Gaussian surface area measures have the uniform upper and lower bound. We
obtain convex bodies' uniform upper and lower bound when $p=0$ in asymmetric
situation and $p\in(0,1)$ in symmetric situation. In fact, for $p\in(0,1)$ ,
there is a counterexample to claim the uniform bound does not exist in
asymmetric situation.
