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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\'einequality. The paper consists of a novel characterization of Poincar\'einequalities and then uses it to give an entirely new proof which is closelyrelated to Muckenhoupt-weights. This new characterization, and the alternativeproof, demonstrate a formal similarity between Muckenhoupt-weights andPoincar\'e inequalities. The proofs we give are short and somewhat more direct.With them we can give the first completely transparent bounds for the quantityof self-improvement and the constants involved. We observe that the quantity ofself-improvement is, for large $p$, directly proportional to $p$, and inverselyproportional to a power of the doubling constant and the constant in thePoincar\'e inequality. The proofs can be localized and thus we obtain moretransparent proofs of the self-improvement of local Poincar\'e inequalities.

Alternative proof of Keith-Zhong self-improvement and connectivity