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Let $X=\{(x,y)\in \mathbb{R}^2| x^2+y^2=1\}$ the unit circle. For each vector $v\in X$ define $H_v=\{w\in \mathbb{R}^2|w\cdot v\geq 0\}$ where $\cdot$ is the dot product. How would someone geometrically describe the family of set $(H_v)_{v\in X}$?

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$H_v$ is a half plane, bounded by the line passing through the origin and perpendicular to $v$. Of these two half-planes, $H_v$ is the one containing $v$.

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$H_v$ is the set of vectors covering the half plane bounded by a line through (1,0) with the slope of the tangent to the unit circle at v.

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  • $\begingroup$ I guess the boundary line passes through the origin, since $v \cdot 0 \geq 0$, hence $0$ must be in $H_v$... $\endgroup$ – Hugo Sep 30 '18 at 22:29

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