Improving transformers ?

The traditional tranformers suffer a lot of losses in various forms, some of them are as follows :
1. Flux leakage
2. Eddy currents
3. Hysteresis

But as we are taught induction we come across the attached figure of two inductors placed coaxially various times. My point is that if we would actually place inductors in this way, we would have the following benefits :

Note : taking inner coil as primary

1. NO NEED OF CORE : As the flux of inner coil passes from outer on its own, no core is needed for transferring flux from primary to secondary
2. NO FLUX LEAKAGE : As entire flux of inner inductor passes from outer, no flux is leaked.
3. NO EDDY CURRENTS: As no core is used no eddy currents are generated.
4. NO HYSTERESIS : Similarly due to absense of core, no loss due to hysteresis is encountered.

So what really is the problem in creating transformers in such a way to reduce losses much more than the current status ?

All relevant opinions and comments appreciated