How the elliptic integrals \(K\) and \(E\) arise from circles and points in the Minkowski plane (Q1331270)

Let \(K\) be a centrally symmetric plane convex body, centered at the origin. Thinking \(K\) as the unit disk for a 2-dimensional Banach space (Minkowski plane), let \(\sigma(K)\) denote the length of \(K\) computed using the metric induced by \(K\). Then \(\sigma(K)\) is called the self- circumference of \(K\). If \(K\) is not necessarily centrally symmetric and \(z\) is any interior point of \(K\), then \(\sigma_ +(K,z)\) and \(\sigma_ -(K,z)\) are defined as the respective positive and negative self- circumferences of \(K\) at \(z\). It is shown that if \(K = B = \) Euclidean unit circle and \(z\) is any interior point of \(B\), \(\sigma_ +(B, z)\) can be expressed in terms of the complete elliptic integral of the second kind. Next the author defines a function \(\tau (B, z)\) that is given by a complete elliptic integral of the first kind. If \(L(K)\) denotes the euclidean length of \(K\) and \(L^*(K)\) that of the polar dual \(K^*\) of \(K\), then the inequality \(L(K)L^*(K) \geq 4 \pi^ 2\) is used. Asymptotic results as \(z\) approaches the boundary of \(B\) are included. An affine transformation is used to discuss similar results for an ellipse.