Needs:

N-Dimensional Line Segments ›

Real Affine Sets ›

Real Subspaces ›

Real Halfspaces ›

Topological Closures ›

Topological Interiors ›

Needed by:

Affine Transformations of Real Convex Sets ›

Convex Real Cones ›

Convex Set Dimensions ›

Convex Sets ›

Real Convex Bodies ›

Real Convex Combinations ›

Real Convex Cones ›

Real Convex Functions ›

Real Convex Hulls ›

Real Convex Optimization Problems ›

Real Convex Sets and Halfspaces ›

Real Extreme Points ›

Real Function Epigraphs ›

Real Polar Sets ›

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Real Convex Sets

Definition

A set $C \subset \R ^n$ is convex if it contains the closed line segment between every pair of points. In the notation of closed line segments, $C$ is convex if

\[ [x, y] \subset C \quad \text{for all } x, y \in C \]

In other words,

\[ \lambda x + (1-\lambda )y \in C \quad \text{for all } x, y \in C \text{ and } \lambda \in [0,1] \]

Roughly speaking, $C$ is convex if and only if its intersection with every line in $\R ^n$ is either empty or a closed line segment.

Examples

The empty set, any singleton, any subspace, any affine set and any halfspace.

Properties

Suppose $\mathcal{K} \subset \powerset{\R ^d}$ is a set of convex sets. Then $\bigcap\mathcal{K} $ is convex.

Suppose $A, B \subset \R ^d$ are convex sets. Then $A + B$, $A - B$ and $\lambda A$ for any real $\lambda $ is convex.

If $A \subset \R ^d$ is convex, then $\cl(A)$ and $\Int(A)$ are convex.¹

Suppose $A \subset \R ^d$ is convex, $x \in A$ and $y \in \Int(a)$. Then all points of the line segement between $x$ and $y$ are members of $\Int(A)$.

Suppose $T: \R ^d \to \R ^d$ is affine. If $A \subset \R ^d$ is convex, then $T(A)$ is convex.

For the first, use $\cl(A) = \bigcap_{\mu > 0} (A + \mu B)$ where $B$ is unit ball of $\R ^d$. ↩︎