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Linear isoperimetric inequality for normal and integral currents in compact subanalytic sets. (English) Zbl 1498.49081

Summary: The isoperimetric inequality for a smooth compact Riemannian manifold \(A\) provides a positive constant \(c\) depending only on \(A\), so that whenever a \(k\)-dimensional integral current \(T\) in \(A\) bounds some integral current \(S\) in \(A\), \(S\) can be chosen to have mass at most \(c\) times the \((k+1)/k\) power of the mass of \(T\). Although such an inequality still holds for any compact Lipschitz neighborhood retract \(A\), it may fail in case \(A\) contains a single polynomial singularity. Here, replacing this power by one, we verify the linear inequality, the mass of \(S\) being bounded by \(c\) times the mass of \(T\), is valid for any compact algebraic, semialgebraic, or even subanalytic set \(A\). In such a set, this linear inequality holds not only for integral currents, which have integer coefficients, but also for normal currents having real coefficients and generally for normal flat chains having coefficients in any complete normed Abelian group. A relative version for a subanalytic pair \((A, B)\) is also true, and there are applications to variational and metric properties of subanalytic sets.

MSC:

49Q15 Geometric measure and integration theory, integral and normal currents in optimization
49Q20 Variational problems in a geometric measure-theoretic setting
32B20 Semi-analytic sets, subanalytic sets, and generalizations
14P10 Semialgebraic sets and related spaces
49J45 Methods involving semicontinuity and convergence; relaxation
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