"Concave" is an adjective describing a surface that curves inward, like the inner surface of a sphere. Its antonym, "convex," describes a surface that curves outward, like the exterior of a sphere. In geometry, concave and convex refer to polygons. A concave polygon has at least one angle greater than 180 degrees. A convex polygon is made of angles each less than or equal to 180 degrees.
"Concave" and "convex" are used to describe gentle, subtle curves, such as those found in mirrors or lenses. Many people find this pair of antonyms similar in sound and thus easy to confuse. An easy way to remember the difference is that a concave indentation in a wall makes a cave, and things that vex someone tend to stick out.
When a mirror is curved into either a concave or convex shape, it produces two opposite illusions. The concave mirror makes the viewer appear tall, while the convex mirror makes the viewer appear short. This is because of a rule that states that the angle of incidence has the same measure as the angle of reflection. The angles in question are between the beam from the viewer's eye and a ray that is perpendicular to the mirror. Concave mirrors make the viewer appear tall because he has to look up to get a perpendicular line: his eyes appear above him. Convex mirrors make the viewer appear short because he has to look down.