Bonnesen's inequality is an inequality relating the length, the area, the radius of the

incircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter ...

and the radius of the

circumcircle In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertex (geometry), vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumrad ...

of a

Jordan curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

. It is a strengthening of the classical

isoperimetric inequality In mathematics, the isoperimetric inequality is a geometric inequality involving the square of the circumference of a closed curve in the plane and the area of a plane region it encloses, as well as its various generalizations. '' Isoperimetric'' ...

. More precisely, consider a planar simple closed curve of length

L

bounding a domain of area

A

. Let

r

and

R

denote the radii of the incircle and the circumcircle. Bonnesen proved the inequality

\pi^2 (R-r)^2 \leq L^2-4\pi A.

The term

L^2-4\pi A

in the right hand side is known as the ''isoperimetric defect''.

Loewner's torus inequality In differential geometry, Loewner's torus inequality is an inequality due to Charles Loewner. It relates the systole and the area of an arbitrary Riemannian metric on the 2-torus. Statement In 1949 Charles Loewner proved that every metric on t ...

with isosystolic defect is a systolic analogue of Bonnesen's inequality.

References

Geometric inequalities {{geometry-stub