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Does anyone know a counterexample to show that a weighted sum of convex sets is not necessarily convex, unless our coefficients are positive?

A weighted sum for me is defined as: $$\alpha C_1 + \beta C_2 = \{ y : y = \alpha x_1 + \beta x_2, x_1 \in C_1, x_2 \in C_2, \alpha , \beta \in \mathbb{R} \}$$.

It is easy to prove this is convex in the case of positive constants, however, the proof does not necessarily take account of the fact that the constants are positive. What is a counterexample of the case for any real coefficients $\alpha$, $\beta$?

Rellek
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1 Answers1

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Weighted sums of convex sets are always convex, even if the factors are negative.

Note that $\alpha C$ is convex for every $\alpha \in \mathbb{R}$, if $C$ is convex. Then use the fact that the Minkowski sum of convex sets is convex.

Dominik
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  • Yep. After all, geometric intuition suggests that the "reflection" of any convex set through the origin should still be convex. – goblin GONE Feb 07 '16 at 22:44
  • I really could not find a counterexample and the proof seems pretty straightforward, but for some reason every source I have seen has stated that weighted sums of nonnegative coefficients preserve convexity. Why make this distinction of the coefficients if it holds in every case? – Rellek Feb 07 '16 at 22:44
  • This probably has something to do with what is considered later in the book. A book on general convex geometry probably doesn't make this distinction, while a book that only considers positive coefficients anyway might do it. – Dominik Feb 07 '16 at 22:53