The \(\epsilon\)-positive center figure (Q1352372)

The area of a circle is usually given as \(\pi R^2\), but it is possibly more illuminating to consider it as \(RC/2\). Any other convex curve does not have a unique radius, but the support function \(H(K,u):S^1\to\mathbb{R}\) with respect to a point \(p\) is one possible generalization. Gage showed, in 1990, that every convex curve \(K\) with length \(L(K)\) and area \(A(K)\) contains at least one point \(p\) such that \(H(K,u) L(K) -A(K)- \pi H(K,u)^2 \geq 0\) for all \(u\in S^1\). Such a point is called a positive center of \(K\). In general such points are not unique. The current author extends the definition, defining an \(\varepsilon\)-positive center to be a point such that \(H(K,u)L(K)- A(K)- \pi H(K,u)^2 \geq \varepsilon\) for all \(u\in S^1\) and some \(\varepsilon>0\). Two examples are shown. Two conjectures are also given: that at least one \(\varepsilon\)-positive center always exists for some \(\varepsilon>0\) whenever the positive center is not unique, and that the set of such points is always convex. (Let \(H(K,u, p)\) be the support function of \(K\), in direction \(u\), with the point \(p\) taken as origin. Then, if \(p= tp_1+ (1-t)p_2\), \(H(K,u,p) =tH(K,u,p_1)+ (1-t)H(K,u,p_2)\). Thus, \(H(K,u,p) L(K)-A(K) -\pi H(K,u,p)^2\) is convex upwards as a function of \(t\), and strictly so when \(u\) is not normal to \(p_1- p_2\). This proves the second conjecture, and appears to make some headway on the first).