I am not an expert and all I can do is pass on what I was told by a mall owner.  I was looking for a place to do a performance and I saw that a local mini-mall had an empty space.  I went to them with your proposition: Rent it to us for less than market and get a tax deduction for the rest.

Not so, they told me.  Here is their explanation:

If they rented it to us for market value (assume $1,000) their income would be $1,000 more and they would pay federal income tax on the $1,000.  They could then give us a check for the $1,000 and deduct the $1,000 leaving them the same place as if they did not rent it to us at all. 

On the other hand, it they rented it to us for nothing, their income would be $1,000 less and they would not pay any federal tax on it since there was no income from it.

They did it anyway - The normal rent was $900 a month and they rented it to us for $100 a month plus two tickets to the show.  It was a slow season and they felt it would not rent until things picked up in the spring.  We both agreed that if it was an emergency - they had a renter who wanted it RIGHT NOW - we would move out.  It worked out well for us but we had to move to another venue for our performances anyway - the store front was too small by the time we had our set in place.

Just my thoughts.