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<div class="csl-entry">Tasso, E. (2025, July 4). <i>Characterizing pure unrectifiability via injectivity of projections</i> [Conference Presentation]. Topics in Geometric Measure Theory, Pisa, Italy. http://hdl.handle.net/20.500.12708/217218</div>
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dc.identifier.uri
http://hdl.handle.net/20.500.12708/217218
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dc.description.abstract
In this talk, we present a geometric characterization of Radon measures in Rn whose orthogonal projections onto (n-1)-planes are singular with respect to the (n-1)-dimensional Hausdorff measure, based on the so-called probabilistic injective projection property. Specifically, we prove that if a Radon measure μ is not supported on a single hyperplane, then it projects singularly with respect to H^(n-1) if and only if a typical orthogonal projection is injective on a set of full μ-measure. As a consequence, we obtain a characterization of codimension-one pure H^(n-1)-unrectifiability within the class of measures that disintegrate atomically with respect to orthogonal projections. In particular, if μ admits a non-trivial absolutely continuous component under projection along a positive measure set of directions, then μ must contain a non-trivial rectifiable part. This result can be viewed as a measure-theoretic analogue of the classical Besicovitch–Federer projection theorem, adapted to the setting of Radon measures. Finally, we show how this framework provides new insight into a longstanding conjecture of Marstrand concerning radial projections of purely unrectifiable sets with finite Hausdorff measure.
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en
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dc.subject
Rectifiability
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Injectivity
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Projections
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Marstrand's Conjecture
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dc.title
Characterizing pure unrectifiability via injectivity of projections
