Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 25I (Differential Geometry)

The periodic Wirtinger inequality says that if $f$ is continuously differentiable, $L$-periodic, and has mean zero, thenEquality holds exactly for a linear combination of $\cos (2\pi s/L)$ and $\sin (2\pi s/L)$.

First suppose $\partial \Omega$ is one positively oriented simple closed curve, parametrized by arc length as $\gamma (s)=(x(s),y(s))$ for $0\le s\le L$. Translating the origin, which changes neither length nor area, makes both $x$ and $y$ have mean zero. Green theorem and the Cauchy-Schwarz inequality giveApplying Wirtinger's inequality to both coordinate functions and using the unit-speed identity $(x^{\prime }{)}^2+(y^{\prime }{)}^2=1$ yieldsand therefore the planar isoperimetric inequalityFor several boundary components, apply the simple-curve result to the relevant enclosed regions and use ${\sum }_j{L}_j^2\le ({\sum }_j{L}_j{)}^2$; holes only decrease the area.

Equality in both inequalities forcesfor constant vectors $A,B$, with the unit-speed and Cauchy-Schwarz equality conditions making $A$ and $B$ perpendicular and equally long. Thus the boundary is a circle. Conversely, a circular domain has area $\pi R^2$ and perimeter $2\pi R$, so equality holds.

Let $A$ and $C$ be opposite interior angles of a convex quadrilateral with side lengths $a,b,c,d$, and let $s=(a+b+c+d)/2$. The Bretschneider formula gives its area $K$ asThe fixed side lengths fix the first term, while the second term is nonpositive. Hencewith equality exactly when $A+C=\pi$. Opposite angles are supplementary exactly when the quadrilateral is a cyclic quadrilateral. The given $Q$ is cyclic, so it attains this upper bound andA crossed or concave competitor can be uncrossed or reflected across a diagonal without changing its side lengths and without decreasing its unsigned area, so the same bound applies. Apart from degenerate coincidences, equality holds precisely when $Q^{\prime }$ is also cyclic with the stated cyclic ordering of its side lengths.