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We extend the definition of the pentagram map from 2D to higher dimensionsand describe its integrability properties for both closed and twisted polygonsby presenting its Lax form. The corresponding continuous limit of the pentagrammap in dimension $d$ is shown to be the $(2,d+1)$-flow of the KdV hierarchy,generalizing the Boussinesq equation in 2D.

The Pentagram map in higher dimensions and KdV flows