Let u be a fixed point in $\displaystyle R^n$ and let c be a fixed real number. Prove that each of the following three sets is convex:

a.

b.

c.

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a. Let

By definition of convex, we claim that the line segment is contained in S.

Since by definition of a scalar product, then

The problem I have is that in R, the set (a) would be a line greater than c, set (b) would be a point, and set (c) would be a line lessthan c.

If in $\displaystyle R^2$, then the set (a) be various regions depending on the signs for ?