Definition

1. Noun:
   • A polygon with all interior angles less than 180 degrees: A convex polygon is a closed, flat shape with straight sides where every internal angle is less than or equal to 180 degrees. This property ensures that all its vertices point outwards.
   • A polygon where any line segment connecting two points inside the shape lies entirely within the shape: This is a key geometric property. For any two points chosen inside a convex polygon, the straight line between them will not cross outside the polygon's boundary.
   • A polygon such that no side extended cuts any other side or vertex: If you extend any of its sides infinitely in a straight line, that line will not intersect or cut through any other side or vertex of the polygon.

Examples of Usage

• Noun:
  • A square and a regular hexagon are classic examples of a convex polygon.
  • In geometry class, we learned to distinguish a convex polygon from a concave one by checking its interior angles.
  • The algorithm only works for the boundary of a convex polygon.

Advanced Usage

• "Strictly convex polygon": A polygon where all interior angles are strictly less than 180 degrees (no angles equal to 180 degrees).
  • A triangle is always a strictly convex polygon.
• In computational geometry: The term is frequently used in algorithms for tasks like collision detection or calculating convex hulls.
  • The "gift-wrapping" algorithm is used to find the convex polygon that encloses a set of points.

Variants and Related Words

• Convex (adj): Curved or rounded outward. The opposite of concave.
  • The lens had a convex surface.
• Convexity (n): The state or quality of being convex.
  • The convexity of the shape makes it structurally strong.
• Convex Hull (n): The smallest convex polygon that can contain a given set of points.
  • The program computed the convex hull of the data points.

Synonyms

• Non-concave polygon: A less common term emphasizing that the polygon is not concave.

Antonyms

• Concave polygon: A polygon with at least one interior angle greater than 180 degrees, where a line segment between two interior points may lie outside the shape.
  • A star shape is often a concave polygon.

Noun

1. a polygon such that no side extended cuts any other side or vertex; it can be cut by a straight line in at most two points