Convex — Definition, Meaning & Examples
Convex
A geometric figure with no indentations. Formally, a geometric figure is convex if every line segment connecting interior points is entirely contained within the figure's interior.
See also
Key Formula
For all points P,Q∈S:tP+(1−t)Q∈Sfor every t∈[0,1]
Where:
- S = The geometric figure (set of points) being tested for convexity
- P,Q = Any two points inside the figure
- t = A parameter ranging from 0 to 1 that traces every point along the line segment from Q to P
- tP+(1−t)Q = A convex combination — any point on the line segment connecting P and Q
Worked Example
Problem: Determine whether the quadrilateral with vertices A(0, 0), B(4, 0), C(4, 3), and D(0, 3) is convex.
Step 1: Identify the shape. The four vertices form a rectangle with width 4 and height 3.
Step 2: Pick two interior points and connect them with a line segment. Choose P(1, 1) and Q(3, 2).
Segment: tP+(1−t)Q=t(1,1)+(1−t)(3,2)
Step 3: Compute a sample point on the segment. At t = 0.5, the midpoint is:
0.5(1,1)+0.5(3,2)=(2,1.5)
Step 4: Check whether (2, 1.5) is inside the rectangle. Since 0 < 2 < 4 and 0 < 1.5 < 3, it is inside.
Step 5: Verify the interior-angle test: compute each interior angle of the rectangle. All four angles equal 90°, and every interior angle is less than 180°. This confirms the polygon is convex.
Each interior angle=90°<180°✓
Answer: The rectangle is convex because every line segment between interior points stays inside the figure, and all interior angles are less than 180°.
Another Example
This example shows how to identify a non-convex (concave) polygon using the cross-product sign test, contrasting with the first example where all angles were under 180°.
Problem: Determine whether the polygon with vertices A(0, 0), B(4, 0), C(2, 1), D(4, 4), and E(0, 4) is convex or concave.
Step 1: Plot the vertices in order. Notice that vertex C(2, 1) pushes inward compared to a regular pentagon shape — this creates an indentation on the bottom edge.
Step 2: Use the interior-angle test at vertex C. Compute the vectors from C to its neighbors B and D.
\vec{CB} = B - C = (4-2,\, 0-1) = (2, -1)$$
$$\vec{CD} = D - C = (4-2,\, 4-1) = (2, 3)
Step 3: Compute the cross product of consecutive edge vectors at C (going around the polygon). The sign of the cross product at C differs from the sign at vertices A, B, D, and E.
BC×CD=(−2)(3)−(1)(2)=−6−2=−8
Step 4: A negative cross product here (while the others are positive for a counterclockwise traversal) means the interior angle at C exceeds 180°. This is a reflex angle, so the polygon is concave.
Interior angle at C>180°⟹not convex
Answer: The polygon is concave (not convex) because the interior angle at vertex C is greater than 180°.
Frequently Asked Questions
What is the difference between convex and concave?
A convex figure has no indentations — all interior angles of a convex polygon are less than 180°, and any line segment between two interior points stays inside. A concave figure has at least one indentation, meaning at least one interior angle exceeds 180° (a reflex angle), and you can find a line segment between two interior points that passes outside the figure.
How do you tell if a polygon is convex?
Check every interior angle: if all are strictly less than 180°, the polygon is convex. Equivalently, compute the cross product of consecutive edge vectors as you traverse the vertices in order. If the cross products all share the same sign (all positive or all negative), the polygon is convex. If any sign differs, it is concave.
Is a circle convex?
Yes. A circle (including its interior) is convex. Any line segment connecting two points inside a circle lies entirely within the circle. In fact, the circle is one of the simplest and most symmetric convex shapes.
Convex vs. Concave
| Convex | Concave | |
|---|---|---|
| Definition | No indentations; every segment between interior points stays inside | At least one indentation; some segments between interior points pass outside |
| Interior angles (polygon) | All interior angles < 180° | At least one interior angle > 180° (reflex angle) |
| Cross-product test | All cross products have the same sign | Cross products have mixed signs |
| Examples | Rectangle, regular hexagon, circle | Star shape, arrow/chevron, L-shaped polygon |
| Diagonals | Every diagonal lies inside the polygon | At least one diagonal passes outside the polygon |
Why It Matters
Convexity appears throughout geometry, optimization, and computer science. In geometry courses, knowing whether a polygon is convex determines which area formulas and angle-sum properties apply directly. In optimization (linear programming), the feasible region is always convex, which guarantees that a local optimum is also a global optimum — a fact that underpins many real-world applications from economics to engineering.
Common Mistakes
Mistake: Confusing a convex polygon with a regular polygon.
Correction: A regular polygon (equal sides and equal angles) is always convex, but a convex polygon does not need equal sides or angles. For example, a non-square rectangle is convex but not regular.
Mistake: Testing only one or two line segments and concluding the shape is convex.
Correction: The convex definition requires that EVERY possible line segment between interior points stays inside. To prove convexity for a polygon, check all interior angles or use the cross-product sign test at every vertex — not just a few sample segments.
Related Terms
- Concave — Opposite property — has at least one indentation
- Geometric Figure — General category that can be convex or concave
- Line Segment — Used in the formal definition of convexity
- Interior — Region that must contain all connecting segments
- Point — Basic element used to test convexity
- Polygon — Common shape tested for convexity
- Regular Polygon — Always convex; a special case