This introduction treats the classical isoperimetric inequality in
Euclidean space and contrasting rough inequalities in noncompact
Riemannian manifolds. In Euclidean space the emphasis is on a most
general form of the inequality sufficiently precise to characterize
the case of equality, and in Riemannian manifolds the emphasis is
on those qualitiative features of the inequality that provide
insight into the coarse geometry at infinity of Riemannian
manifolds. The treatment in Euclidean space features a number of
proofs of the classical inequality in increasing generality,
providing in the process a transition from the methods of classical
differential geometry to those of modern geometric measure theory;
and the treatment in Riemannian manifolds features discretization
techniques, and applications to upper bounds of large time heat
diffusion in Riemannian manifolds. The result is an introduction to
the rich tapestry of ideas and techniques of isoperimetric
inequalities, a subject that has its beginnings in classical
antiquity and which continues to inspire fresh ideas in geometry
and analysis to this very day--and beyond
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