Inequalities that Imply the Isoperimetric Inequality

The isoperimetric inequality says that the area of any region in the plane bounded by a curve of a fixed length can never exceed the area of a circle whose boundary has that length. Moreover, if some region has the same length and area as some circle, then it must be the circle. There are dozens of proofs. We give several arguments which depend on more primitive geometric and analytic inequalities. “The cicrle is the most simple, and the most perfect figure. ” Poculus. Commentary on the first book of Euclid’s Elements. “Lo cerchio è perfettissima figura.” Dante. In these notes, I will present a few of my favorite proofs of the isoperimetric inequality. It is amusing and very instructive to see that many different ideas can be used to establish the same statement. I will concentrate on proofs based on more primitive inequalities. Several important proofs are omitted (using the calculus of variations, integral geometry or Steiner symmetrization), due to the fact that they have already been discussed by others at previous occasions [B], [BL], [S]. Among all closed curve of length L in the plane, how large can the enclosed area be? Since the regions are enclosed by the curves with the same perimeter, we are asking for the largest area amongst isoperiometric regions. Which curve (or curves) encloses the largest possible area? We could ask the dual question: Among all regions in the plane with prescribed area A, at least how long should the perimiter be? Since the regions all have the same area, we are asking for the smallest length amongst isopiphenic regions. Which figures realize the least perimeter? The answer to both questions is the circle. A way to formulate a statement equivalent to both is in the form of the Isoperimetric Inequality. 1 2 Theorem. Isoperimetric Inequality. Among all regions in the plane, enclosed by a piecewise C boundary curve, with area A and perimeter L, 4πA ≤ L. If equality holds, then the region is a circle. One doesn’t need to assume this much smoothness on the boundary for the isoperimetric inequality to hold, rectifiability suffices [B],[G]. This is for simplicity. The student may be more familiar with rigorous arguments under such hypotheses. 1. Some applications of the isoperimetric inequality. This is a useful inequality. Following Polya[P] let’s illustrate how it may be used to prove things that may be much trickier to prove using calculus alone. The largest quadrilateral problem. Suppose four side lengths a, b, c, d are given. Which quadrilateral in the plane with these side lengths maximizes the area? The answer is the cyclic quadrilateral, the one whose vertices can be placed in order around a circle, i.e. the one inscribable in some circle. First of all, can a quadrilateral whose side lengths are four numbers be consrtructed at all? The answer is yes provided that the side lengths satisfy the inequalities that require that the sum of the lengths of any three of the sides exceeds the length the remaining one. (See Lemma 1. of the Appendix.) One can also prove that there is a cyclic quadrilateral with side lengths a, b, c, d provided that the inequalities are strict [B]. (See Lemma 2. of the Appendix.) Fig. 1. Suppose we wished to show that among quadrilaterals in the plane with given four side lengths a, b, c, d, that the cyclic quadrilateral has the largest area. This problem can be handled by calculus, but it may become messy depending on your formulation. We will show this fact independently of the isoperimetric inequality later in two ways, using trigonometry (Brahmagupta’s Inequality) and as a consequence of Ptolemy’s inequality. After all we will wish to deduce the isoperimetric 3 inequality from these! But assuming the isoperimetric inequality gives an easiest argument that cyclic quadrilaterals have the largest area. By Lemma 2. we can construct a cyclic quadrilateral with the given side lengths. Let ABCD be its vertices in order on Z, the circle containing the vertices. Now imagine the regions between the sides of the quadrilateral and the circle are made of a rigid material such as a stiff plastic. Furthermore, imagine that the quadrilateral with its plastic flanges attached has hinges at the vertices, so that the entire figure can flex in the plane. The problem of finding the largest area enclosed amongst these flexings is the same as finding the largest area enclosed by the quadrilateral is the same as finding the largest area enclosed by the the circular arcs, because the additional plastic includes a fixed area. By the isoperimetric inequality, the circular figure has the largest area, thus the cyclic quadrilateral contains the largest area. This argument applies to polygons with any number of sides. Hence, the polygon with given side length has largest area if and only if the polygon is inscribed in a circle. 2. Princess Dido’s Problem. Here is another application we shall motivate by a tale from antiquity [P]. Princess Dido, daughter of a Tyrian king and future founder of Carthage purchased from the North African natives an amount of land along the coastline “not larger than what an oxhide can surround.” She cut the oxhide into strips and made a very long string of length L. And then she faced the geometrical problem of finding the region of maximal area enclosed by a curve, given that she is allowed to use the shoreline as part of the region boundary. In the interior of the continent the answer would be the circle, but on the seashore the problem is different. Assuming that the seashore is a straight line (the x-axis), then the maximum area enclosed is a semicircle with its diameter on the shoreline and area A = L/π. This easily follows from the isoperimetric inequality. We regard the shoreline as a mirror and reflect the curve Γ whose points are (x(s), y(s)) to Γ whose corresponding points are (x(s),−y(s)) on the other side of the axis. Then the composite curve Γ ∪ Γ has total length 2L. The maximal area enclosed is the circle. Centering the circle on the shoreline means that the upper and lower semicircles coincide under reflection, thus the lower semicircle is the maximal area region. Exercise. Suppose Dido bargained to buy land along a cape (in {(x, y) : y ≤ −c|x|}, for c > 0 some constant.) What now would be the shape of the largest land she could surround with a string of length L? (see [P].) 3. Ptolemy’s Inequality, Complex Numbers and the Quadrilateral Inequality. We envision a theorem about the four vertices of a quadrilateral, but the statement holds for any four points. Theorem. Ptolemy’s Inequality. Let ABCD be four points in the plane. Let a, b, c, d be the lengths of the sides AB, BC, CD, DA and p, q be the lengths of the diagonals AC, BD, resp. Then (1) pq ≤ ac + bd. 4 If equality holds, then ABCD is cyclic (its vertices lie in order on a circle) or it’s contained in a line segment (the length of one side equals the sum of the other three.)