I have a question which I believe to be pretty basic.

Let $\Gamma$ be some convex body, bounded inside a $L_2$ sphere of radius 1 $B(0,1)$.

Is it true that the surface area of $\Gamma$ is smaller than the surface area of the sphere?

I'm guessing that the answer involves finding a continuous deformation from $\Gamma$ to the sphere for which the area is monotonous, but I'm incapable of finding it