A Convex Polygon Is the Intersection of Half Planes

The applet below is a tool that helps verify that the intersection of any (finite) number of half planes constitutes a convex polygon, which might be empty, too. Lines in the applet can be moved by pointing and dragging the cursor. If you point close to the border of the drawing area, the lines will rotate around the opposite point. Otherwise, they will translate preserving the orientation.

In fact, the intersection of any number of convex shapes is again convex. This follows directly from the definition of convexity. For a finite number of shapes there is also a simple proof by induction.

(The applet requires Java 2, jre 1.4.2 or higher. Check with the Sun's site.)

Convex Sets

• Helly's Theorem
• First Applications of Helly's Theorem
• Crossed-Lines Construction of Shapes of Constant Width
• Shapes of constant width (An Interactive Gizmo)
• Star Construction of Shapes of Constant Width
• Convex Polygon Is the Intersection of Half Planes
• Minkowski's addition of convex shapes
• The Theorem of Barbier
• A. Soifer's Book, P. Erdos' Conjecture, B. Grunbaum's Counterexample

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Copyright © 1996-2012 Alexander Bogomolny