It may be a little uncommon to post an answer to one's own question, but for a good record-keeping I think it is worth mentioning that a complete and very explicit answer is now known. The function in question is given by $$ F(x) = \min \{ k\|x\|^{1-1/k}\colon k\ge 1 \}, $$ where $k$ runs over all positive integers, and $\|x\|=\min\{x,1-x\}$. The graph of this function looks like this:

the graph http://math.haifa.ac.il/%7Eseva/MathOverflow/F.jpg

The proof and applications can be found here.

It may be a little uncommon to post an answer to one's own question, but for a good record-keeping I think it is worth mentioning that a complete and very explicit answer is now known. The function in question is given by $$ F(x) = \min \{ k\|x\|^{1-1/k}\colon k\ge 1 \}, $$ where $k$ runs over all positive integers, and $\|x\|=\min\{x,1-x\}$. The graph of this function looks like this:

(source)

The proof and applications can be found here.