Give an example of two convex sets A, B ∈R^2 for which cl A+cl B ⊂cl(A+B) but cl A+cl B ≠cl(A+B)

step 1 We're asked to prove that the sum of convex sets is again a convex set. So let A and B be convex

Give an example of two convex sets A, B ∈R^2 for which cl...

Give an example of two convex sets $A, B \in \mathrm{R}^2$ for which $\mathrm{cl} A+\mathrm{cl} B \subset$ $\operatorname{cl}(A+B)$ but $\operatorname{cl} A+\operatorname{cl} B \neq \operatorname{cl}(A+B)$. (Hint: Since $A$ and $B$ are convex, you may find it convenient to replace $\operatorname{cl} A, \mathrm{cl} B$, and $\operatorname{cl}(A+B)$ by their equivalent linear counterparts: $\operatorname{lcl} A, \operatorname{lcl} B$, and $\operatorname{lcl}(A+B)$.)

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Key Concepts

Set Closure

The closure of a set is the smallest closed set that contains the original set, meaning it includes all the limit points of the set. This concept is crucial in topology, ensuring that operations such as limits and convergences are well-behaved within the space.

Minkowski Sum

The Minkowski sum of two sets is the set obtained by adding each element of one set to each element of the other. This operation is particularly important in convex analysis and geometry, as the sum of two convex sets is itself convex, and it plays a role in optimization and functional analysis.

Convex Sets

A convex set is one in which, for any two points inside the set, the entire line segment connecting them is also contained within the set. This property is fundamental in many areas of mathematics and optimization because it implies that the set has no 'holes' or 'indentations', making analysis and computation more tractable.

Interaction of Closure and Minkowski Sum

When combining operations such as closure and Minkowski addition, the order in which these operations are performed can affect the result. In some cases, the sum of the closures of two convex sets can be strictly contained in the closure of their Minkowski sum, highlighting subtle details about convergence and accumulation points in the combined set.

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