Joseph B. Keller and Ravi Vakil. Department of Mathematics. Stanford University, Stanford, CA February 15, 2008

Transcription

1 π p, the value of π in l p Joseph B. Keller and Ravi Vakil Department of Mathematics Stanford University, Stanford, CA February 15, 8 The two dimensional space l p is the set of points in the plane, with the distance between two points x, y) and x, y ) defined by x x p + y y p ) 1/p, 1 p. The distance of x, y) from the origin is then x p + y p ) 1/p. The equation of the unit circle C p, i.e. the circle with its center at the origin and radius 1, is x p + y p ) 1/p = 1. 1) Figure 1 shows C p for p = 1, 3/,, 3 and. Equation 1) is unchanged when x is replaced by x, when y is replaced by y, and when x and y are interchanged. Therefore C p is symmetric about the y-axis, about the x-axis, and about the line x = y. 1

2 1. 1. p = 8 y.8.6 p = 1 p = p = 3/ p = x Figure 1. The unit circle C p in the first quadrant, defined by 1), for p = 1, 3/,, 3,. It is natural to define π p as the ratio of the circumference of C p in the p-metric) to two times its radius also in the p-metric), which is its diameter,. This definition has been well studied, see for example [], [1], and [3]. The circumference is the integral of the element of arclength ds = dx p + dy p ) 1/p around C p. Thus π p = 1 dx p + dy p ) 1/p = 1 C p C p dy 1 + dx p ) 1/p dx. ) Because of the symmetry of C p, its circumference is equal to four times its arclength in the first quadrant, or eight times its arclength in the first quadrant between the lines x = and x = y. When x = y, 1) shows that x = 1/p, so the integral in ) is 8 times the integral from to 1/p. By calculating dy/dx from 1), we can rewrite ) as π p = 4 1/p [ 1 + x p 1 1 p] 1/p dx. 3) For p = 1, 3) yields π 1 = 4 1/p ) 1/p ) = 4. For p =, the integrand is 1 and the upper limit is 1, so π = 4. At p =, π = π. If geometry had been developed using the l p distance instead of the l distance, π p would have replaced π, which is just the familiar π.

3 Figure shows a graph of π p as a function of p, obtained by numerical integration of 3). The graph suggests that as p increases from p = 1, π p decreases monotonically from its maximum value π 1 = 4 to its minimum value π = π, and then increases monotonically to π = 4. In fact this is the case, and was proved by Adler and Tanton in [1]. Thus for each p in 1 p, there is a q in q such that Π p p π p = π q. 4) Figure. π p as a function of p, obtained by numerical integration of 3). To find q we recall that the Hölder inequality involves two exponents p and q related by 1 p + 1 q = 1. 5) The numerical results shown in Figure lead one to conjecture that 4) will hold when 5) does. Indeed, Adler and Tanton earlier asked precisely this question as the concluding remark to [1]. In fact, when 5) holds, the domains bounded by C p and C q are polar to one another. Then a result of Schäffer [4] see also Thompson [5]) shows that 4) holds. This argument from Minkowski geometry suggested to us that there should be a direct elementary explanation. 3

4 We shall now give another proof that 4) holds when 5) does, by showing that then the integral 3) for π p equals that for π q. We begin by writing the equation for the arc of C p in the first quadrant in terms of a parameter t [, ], setting x = f 1 t) and y = f t). Then the length L p of that arc can be written as L p = f pp 1) f 1 + f pp 1) ) 1/p f p 1 1 dt = f 1 + f f 1 ) pp 1) 1/p dt. 6) The integrand in 6) is obtained from that on the right side of ). We choose the parameter t such that t q/p is the slope of the line from the origin to the point f 1 t), f t) on C p, so that t q/p = f t)/f 1 t). From this equation and 1) we find that f 1 t) = t q + 1) 1/p, f t) = t q + 1 ) 1/p. 7) We parameterize C q in the same way, setting x = g 1 t) and y = g t), with g 1 t) = t p + 1) 1/q, g t) = t p + 1 ) 1/q. 8) Now we define the function F t) = f 1 g + f g 1. At the ends of the two arcs, t = and t =, we have f 1 = g 1 and f = g. Therefore F ) = and F ) =, so F t)dt =. This equation can be rewritten as follows, by differentiating the definition of F t) to get F t) and then transposing: f 1g + f g 1 ) dt = g 1f + g f 1 ) dt. 9) The integrand on the left side of 9) can be rewritten as f 1 g + f g 1 = f ) pp 1) f 1 + f 1 4 1/p. 1)

5 To prove 3) we first transform the left side as follows: ) f 1g + f g 1 = 1 p ) + 1 p t q + 1) p+1 p qt q 1 t p + 1) 1/q t q + 1) p+1 p q)t q+1) t p + 1) 1/q p t p + 1) 1/q t q 1+p/q + t qp+1)/p q+1) ) p t p + 1) 1/p 1 t 1 p 1 1 t p + 1) p t p + 1) 1/p t 1 p ) Then we transform the right side: f ) pp 1) f 1 + f 1 1/p = f 1 + tq/p) pp 1) ) 1/p = f 1 + t p ) 1/p = 1 p t q + 1) p+1)/p q)t q t p ) 1/p p t p + 1) 1/p t q p p+1) q 1 p t p + 1) 1/p t 1 p ) The last forms of 11) and 1) are the same, which proves 1). The integral from to of the right side of 1) is just L p, as 6) shows. Therefore the integral of the left side, which is also the left side of 9), is also L p. A symmetrical argument shows that the right side of 9) is L q, so 9) shows that L p = L q. This proves that 4) is true when p and q are related by 5). This argument is not just formal manipulation. It is geometrically motivated, and the geometry behind this proof is rather interesting. Suppose q > p. As t goes from to 1, the point on C p is behind the point on C q. At t = 1 the p-point passes the q-point. 5

6 The cumulative lengths at time t are not the same; the difference is twice the area of the triangle spanned by the origin, the p-point and the q-point. This difference vanishes at t =, when the two points coincide. Acknowledgement Jonathan C. Mattingly and Arnold D. Kim calculated π p. Rafe Mazzeo and Zhongmin Shen brought to our attention the work of Schäffer. We thank them all. We also thank the referees for bring to our attention many references we were not earlier aware of, and for comments which substantially improved the exposition of this article. References [1] Adler, C.L. and Tanton, J., π is the minimum value of Pi, The College Mathematics Journal 31, 1 16,. [] Euler, R. and Sadek, J., The πs go full circle, Mathematics Magazine 7, 59 63, [3] Poodiack, R., Generalizing π, angle measure, and trigonometry, Aug. 3, 4, [4] Schäffer, J.J., The self-circumference of polar convex bodies, Arch. Math. 4, 87 9,

7 [5] Thompson, A.C., Minkowski Geometry, Cambridge Univ. Press 1996, p. 118, Corollary and p., Corollary