Are there no convection currents in ice-boxes and aerogel? Technologies dealt with
Ice-boxes have been used for a long time to prevent ice from melting for a longer time than keeping it in open. Aerogel is still an emerging technology (I came across it on YouTube), and is a promising insulator.
My Thoughts 
I thought that since the inner wall of the ice-box is close to the ice, it would be cooler than the outer wall which would be warmer.
Since there is temperature difference between the two walls, thereby temperature difference between layers of air, shouldn't there be convection currents?
But then, I also think that the space is pretty crammed, I mean the one filled by the air, so there wouldn't be “layers” of air, hence no such convection currents which could cause significant loss of heat from the ice box.
P.S. The question does not totally deny that there are absolutely no convection currents, but just some molecules (significantly less compared to the total number of molecules trapped in the ice-box, dunno about the aerogel).
I have no ideas for the aerogel, you know, the pores are small to the molecular level, so I am not sure about it.
The Question-Revisited
Are there convection currents in the ice-box and aerogel, I mean, layers of air moving (for the ice-box), and air molecules changing position or moving through the pores for aerogel when heated form one side?
 A: There are a few ways this can play out.
Wikipedia suggests that the structure of an aerogel actually greatly reduces even the conductivity of the gas; by restricting it's movement to near mean free path length for gas molecules.  Another consequence of this is that it cannot transfer heat through convection at all.  (more on Knudsen effect)
For icebox walls, there are two ways it can be done.  Generally, the walls would have some sort of insulation between the inner and outer surfaces, to minimize the rate of conduction between the two.  Insulation is generally made up of a porous substance which allows for conductive heat transfer through the trapped air; but does not offer enough space for effective convection currents; and thus the effect of heat transfer due to convection is greatly reduced.  
If the walls were hollow; but the space is thin enough, there is still a chance that the space is not adequate to allow sufficient convection currents.  This would be similar to the behaviour of insulated glass, where the panes are too close together to allow for significant convective currents to develop; thus causing the air sandwiched between to act as a layer of insulation.  Typically for something like an icebox though, I would expect to see insulation in the walls, since unlike glass, you don't need to be able to see through it while still minimizing heat transfer.
