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Proof that the Intersection of Convex Sets is Convex

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The Formal Theorem

Let {Ci}i∈I \{ C_i \}_{i \in I} be an arbitrary collection of convex sets in a real vector space V V . The intersection C=⋂i∈ICi C = \bigcap_{i \in I} C_i is convex. Formally, for any two points x,y∈C x, y \in C and any scalar λ∈[0,1] \lambda \in [0, 1] , the convex combination must satisfy:
(1−λ)x+λy∈C (1 - \lambda)x + \lambda y \in C

Analytical Intuition.

Imagine the vast, multidimensional landscape of your vector space V V . Each set Ci C_i represents a 'safe zone'—a region where, if you pick any two points, the entire straight line connecting them remains safely inside that zone. Now, consider the intersection C C , the 'common ground' shared by every single one of these zones. If you take two points x x and y y that reside in this common ground, they must—by definition—exist in every individual set Ci C_i . Because every Ci C_i is convex, the line segment connecting x x and y y is guaranteed to exist within each individual Ci C_i . Since this segment is present in every single set simultaneously, it must inherently exist in their intersection C C . Convexity is a structural 'inheritance' property: when multiple constraints define a region, the resulting geometry preserves the straight-line integrity of its constituents. Whether you are intersecting two circles or an infinite family of hyperplanes, the internal consistency of the 'straight path' remains unbroken.
CAUTION

Institutional Warning.

Students often struggle with the indexing set I I . They mistakenly assume this only applies to finite intersections. However, the definition of convexity holds regardless of whether I I is finite, countable, or uncountable, because the property must hold for each set individually.

Academic Inquiries.

01

Does this property hold for unions as well?

No. The union of two convex sets is generally not convex. Think of two disjoint disks; a line segment connecting a point in the first to a point in the second will pass through the space between them, which is outside the union.

02

What if the intersection is empty?

The empty set is vacuously convex, as there are no pairs of points x,y x, y that violate the condition.

Standardized References.

  • Definitive Institutional SourceBoyd, S., & Vandenberghe, L., Convex Optimization

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Proof that the Intersection of Convex Sets is Convex: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/fundamentals-of-optimization/proof-that-the-intersection-of-convex-sets-is-convex

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