Abstract
In Chapter 4, we proved the Poincare inequality in ℝ n by using the fact that points in ℝ n can be joined by a thick “pencil” of curves. In the previous chapter, we defined the Loewner function of a metric space that detects quantitatively whether or not the space contains rectifiable curves. In this chapter, we will see that these two concepts, the Poincare inequality and the Loewner condition, are related.
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