Convex Polygon
A convex polygon has all interior angles less than 180 degrees, with every diagonal lying entirely inside the figure.
Definition
A convex polygon has no "dents" or inward-pointing corners, every angle points outward, and all interior angles measure less than $180^\circ$; equivalently, every line segment connecting two interior points (and every diagonal) lies entirely inside the polygon. All regular polygons are convex, the exterior angle sum is exactly $360^\circ$, and formally, a polygon $P$ is convex if and only if it is the intersection of half-planes; its vertices are exactly the extreme points of its convex hull.
Example
A square, triangle, regular hexagon, and stop sign are all convex; an arrow or star shape has dents and is not convex, always looking like it "bulges out" instead. A regular pentagon is convex since all interior angles are $108^\circ<180^\circ$; an arrowhead with one reflex interior angle greater than $180^\circ$ is concave. Given points $(0,0), (1,0), (0,1), (0.5,0.5)$, the convex hull is the triangle $(0,0), (1,0), (0,1)$, since $(0.5,0.5)$ lies inside, findable in $O(n\log n)$ time by algorithms like Graham scan or Jarvis march.
Key Insight
A simple test: if a rubber band stretched around the shape touches every vertex, it is convex; if the band "skips over" any vertex because it is pushed inward, the polygon is concave. Convex polygons have many properties lost for concave ones: all diagonals stay inside, triangulation is easy, the centroid is always interior, and convexity is the key assumption behind the fundamental theorem of linear programming (the simplex method), where a function minimized over a convex polygon attains its minimum at a vertex, a principle that generalizes to convex polytopes in higher dimensions.