An object is **convex** if any point lying directly between two points of the object is also in the object. For example, a solid cube is convex, but anything that is hollow or has a dent in it is not convex.

A real-valued function*f* defined on an interval (or on any convex set) is called **convex** if for any two points *x* and *y* in its domain and any *t* in [0,1], we have

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In mathematics, convexity can be defined for subsets of any real or complex vector space. Such a subset *C* is said to be **convex** if, for all *x* and *y* in *C* and all *t* in the interval [0,1], the point *tx* + (1`-`*t*)*y* is in *C*.

The convex subsets of **R** (the set of real numbers) are simply the intervals of **R**.
Some examples of convex subsets of Euclidean 3-space are the Archimedean solids and the Platonic solids.
The Kepler solids are examples of non-convex sets.

The intersection of any collection of convex sets is itself convex, so the convex subsets of a (real or complex) vector space form a complete lattice. This also means that any subset *A* of the vector space is contained within a smallest convex set (called the **convex hull** of *A*), namely the intersection of all convex sets containing *A*.

A real-valued function

*f*(*tx*+ (1`-`*t*)*y*) ≤*t**f*(*x*) + (1`-`*t*)*f*(*y*).

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