Post #3179063

@tao@mathstodon.xyz Start at a seam with F=0 and \alpha_{\rm in}<0. Enforce \alpha’(s)\equiv -1 while \alpha<0. This gives F’\equiv 0, so we track the barrier r=\cos\alpha and obtain \kappa_{\mathrm{par}}=\kappa_{\mathrm{mer}}=1. After \alpha=0, take any C^\infty profile with -1\le \alpha’\le 1. Then \sin\alpha\ge 0 and F’\ge 0, hence F\ge 0. Therefore \kappa_{\mathrm{par}}\le 1 and \kappa_{\mathrm{mer}}\le 1 across the blend. If the genus 2 bowl volume is about 1.4, then two corks add about 1.17. Total is about 2.57<4\pi/3. This suggests a genus 0 body with |\kappa_i|\le 1 can beat the unit ball once convexity is dropped. This isn’t fully fleshed out sorry!!!