  We give a new proof for the self-improvement of uniform p-fatness in the
setting of general metric spaces. Our proof is based on rather standard methods
of geometric analysis, and in particular the proof avoids the use of deep
results from potential theory and analysis on metric spaces that have been
indispensable in the previous proofs of the self-improvement. A key ingredient
in the proof is a self-improvement property for local Hardy inequalities.
