Let $X$ and $Y$ be two convex cones, and denote by $(1/2)*X+(1/2)*Y$ the Minkowski average of $X$ and $Y$ (i.e., $\{z:z=(1/2)*x+(1/2)*y,x\in X,y\in Y\}$). Then $$X\cap Y \subseteq(1/2)*X+(1/2)*Y$$ Is this claim true?
2026-04-09 05:56:55.1775714215
Is the following convex geometry relating intersection and set averages true?
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Take $z\in X\cap Y$. Then $z=(1/2)*z+(1/2)*z$ and therefore $z\in(1/2)*X+(1/2)*Y$.