04 - Projections, I

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What do the shadows of sets look like? More precisely, what happens to a set under orthogonal projections? This week we study the effects of projections on Hausdorff measure when we project into almost every plane. We first begin by making sense of what we mean by ?almost every plane? by introducing the Grasmannian and its associated measure. Next, we introduce energies, a convenient object for estimating dimension that play well with projections.