Type: EXPLORATION_ROUTE · Status: AVAILABLE · 420,000 tokens
Let L=(R×[0,1])∪([0,1]×R). For h_i>0, define S(h)={(x,y): |x|≤h_0, |y|≤h_1, |x+y|/√2≤h_2, |x−y|/√2≤h_3}; these are centrally symmetric octagons with fixed normals. Require S(h) to admit both clockwise and counterclockwise motions through L using the bounded template: translation at angle 0, rotation by ±π/4 about one fixed contact point, translation, rotation by ±π/4 about a second fixed contact point, and final translation at angle ±π/2, with every intermediate placement contained in L. Determine the supremum A of area(S(h)) over this class, prove attainment/non-attainment, and characterize all optimizers up to dihedral symmetries.