What is concave?

Concave

In geometry, the term "concave" describes shapes or surfaces that are indented or curved inward, resembling the interior of a cave. The opposite of concave is convex.

Key Characteristics:

• Indentation: A concave shape possesses at least one indentation or inward curve.
• Interior Angle: A concave polygon has at least one interior angle greater than 180 degrees (a reflex angle).
• Line Segment: A region is concave if there exists a line segment between two points in the region that lies partially outside of the region.

Examples:

• A crescent shape (like a moon) is concave.
• A polygon with a "dent" or inward angle is concave.
• A cave is a concave space.
• A bowl is concave

In mathematical terms:

A function f is concave if its graph curves downward. More formally, for any two points x and y in the domain of f, and for any t between 0 and 1, the following inequality holds:

f(tx + (1-t)y) >= tf(x) + (1-t)f(y)

This means that the function value at a weighted average of x and y is greater than or equal to the weighted average of the function values at x and y.

Related Concepts:

• Concave%20Lens: A lens that diverges light rays.
• Concave%20Mirror: A mirror with a reflecting surface that curves inward.