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You haven't explained what you mean by circle. Let me consider three possible interpretations: The set of points at distance exactly $r$ from the origin, for some $r$. The set of points at distance at most $r$ from the origin, for some $r$. The set of points at distance less than $r$ from the origin, for some $r$. In all cases, it is easy to see that the ...


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I am using the definition of pseudo-dimension found here, page 10 of the pdf. Denote $h_1(x) = 2x+1$, $h_2(x) = 4x$, $h_3(x) = x$ and $h_4(x) =3x+4$. Let's consider $C = (-2, 2)$ a vector. Then $r = (-2.5, 6)$ is a witness that proves that $C$ is pseudo-shattered by $\mathcal{H}$: $h_1(-2) = -3 <-2.5$, $h_1(2) = 5 <6$; $h_2(-2) = -8 <-2.5$, $h_2(2) ...


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