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Prove set $S$ is convex. $$S = \{x \in R^2 \mid v^{T}x \geq a\}$$

Can someone explain to me how to prove if this set is convex or not? Are inner products always convex? I know that to prove the set is convex is to show that for all $x,y \in C$ and all $\alpha \in [0,1]$, $\alpha x + (1- \alpha) \in C$.

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Guide:

Suppose $x_1, x_2 \in S$, that is $v^Tx_i \ge a, i = 1, 2$.

Verify that $v^T(\alpha x_1 + (1-\alpha)x_2) \ge a$.

If you want to check the result for other inner product, rather than working with $v^Tx$, work with $\langle v, x \rangle $ instead.

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