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Which of the following statements is true ?

  • A.Every connected subset of \(\mathbf{R}^{n}\) is convex
  • B.Every convex set in \(\mathbf{R}^{n}\) is compact
  • C.Every convex subset of \(\mathbf{R}^{n}\) is connected
  • D.The set : \(\{(x, y) \in \mathbf{R}^{2} / x \ge 0, y \ge 0\}\) \(\cup\ \{(x, y) \in \mathbf{R}^{2} / x \le 0, y \le 0\}\) is convex

Answer: C

The true statement is Every convex subset of Rn is connected.

A convex set contains the segment between any two of its points, so it is path connected, and path connected sets are connected.

The converse fails, which rules out the first option: a circle, or an annulus, is connected but not convex. Convex sets need not be compact: Rn itself and any open ball are convex but not compact. The union of the closed first and third quadrants is connected (both contain the origin) but not convex: the points (1, 0) and (0, -1) lie in it while their midpoint (1/2, -1/2) lies in the fourth quadrant.

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