  We find a new proof for the celebrated theorem of Keith and Zhong that a
$(1,p)$-Poincar\'e inequality self-improves to a $(1,p-\epsilon)$-Poincar\'e
inequality. The paper consists of a novel characterization of Poincar\'e
inequalities and then uses it to give an entirely new proof which is closely
related to Muckenhoupt-weights. This new characterization, and the alternative
proof, demonstrate a formal similarity between Muckenhoupt-weights and
Poincar\'e inequalities. The proofs we give are short and somewhat more direct.
With them we can give the first completely transparent bounds for the quantity
of self-improvement and the constants involved. We observe that the quantity of
self-improvement is, for large $p$, directly proportional to $p$, and inversely
proportional to a power of the doubling constant and the constant in the
Poincar\'e inequality. The proofs can be localized and thus we obtain more
transparent proofs of the self-improvement of local Poincar\'e inequalities.
