Fenchel's theorem
Encyclopedia
In differential geometry, Fenchel's theorem (Werner Fenchel
, 1929) states that the average curvature
of any closed convex plane curve is
where P is the perimeter. More generally, for an arbitrary closed curve in space the average curvature is
with equality holding only for convex plane curves.
Werner Fenchel
Moritz Werner Fenchel was a mathematician known for his contributions to geometry and to optimization theory. Fenchel established the basic results of convex analysis and nonlinear optimization theory. Fenchel's monographs and lecture-notes were very influential also...
, 1929) states that the average curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
of any closed convex plane curve is
where P is the perimeter. More generally, for an arbitrary closed curve in space the average curvature is
with equality holding only for convex plane curves.