Geodesic
PCA sets and convexity
Fundamenta Mathematicae, Tome 163 (2000) no. 3, pp. 267-275
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Three sets occurring in functional analysis are shown to be of class PCA (also called $Σ^1_2$) and to be exactly of that class. The definition of each set is close to the usual objects of modern analysis, but some subtlety causes the sets to have a greater complexity than expected. Recent work in a similar direction is in [1, 2, 10, 11, 12]. 
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Robert Kaufman. PCA sets and convexity. Fundamenta Mathematicae, Tome 163 (2000) no. 3, pp. 267-275. doi: 10.4064/fm-163-3-267-275

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