Instead of glucose or glycerol, is it possible for E coli to grow on an amino acid mixture as the sole carbon source and, if so, what should the concentration be?
E.coli can do that and in fact does this a lot in a commonly used bacterial medium: The popular LB Medium. This is composed of three components:
Tryptone, a peptide mix made by digesting milk casein with the peptidase Trypsin (10g/l)
Yeast extract, made by autolysis of the cells (5g/l)
Sodium chloride (10g/l)
The original formulation of Bertani from 1951 (see reference 1) also contains 1g/l Glucose, which was later discarded, since the handling is easier without it (glucose cannot be autoclaved as it degrades).
Since the medium does not contain any other utilizable sugars as a carbon source, the bacteria use free amino acids (and small peptides up to 650 Da in size) for their growth. The reason for this limit is the exclusion size of the bacterial porin channels which allows the uptake of the peptides and free amino acids (see reference 2 for details).
Once these are exhausted, the growth stops, which is the reason why LB Medium is actually not very good suited to get really high densities (that's why SOC media are used for some applications). Read the excellent paper in reference 3 and the also excellent article in reference 4 if you want to know more about the topic.
I don't think it matters from which source your amino acids are coming. Regarding the concentration I haven't found any data, but in reference 4 it is estimated that only about 25% can be used by the bacteria. Since the mixture contains about 15 grams of protein lysate, this would roughly mean about 4 grams of free amino acids or more.