{"version":10,"randomSeed":"3e507318ddd9294515610556195f89f1","graph":{"viewport":{"xmin":-2.629688357253654,"ymin":-2.1068565191164295,"xmax":3.0265433592288122,"ymax":2.728829980636423}},"expressions":{"list":[{"type":"text","id":"2","text":"inspired by \"The Nazarov proof of the non-symmetric Bourgainâ€“Milman inequality\" by Vlassis Mastrantonis and Yanir Rubinstein (2022) https://arxiv.org/abs/2206.06188"},{"type":"expression","id":"3","color":"#2d70b3","latex":"K\\left(x,y\\right)=2y\\cdot\\frac{\\left(\\cos\\left(2\\pi x\\right),\\sin\\left(2\\pi x\\right)\\right)}{\\cos\\left(2\\pi\\left(x-\\frac{1}{6}\\operatorname{round}\\left(6x\\right)\\right)\\right)}"},{"type":"expression","id":"4","color":"#388c46","latex":"N=100","slider":{"hardMin":true,"hardMax":true,"min":"1","max":"1000","step":"1"}},{"type":"text","id":"12","text":"approximation of the Convex Body"},{"type":"expression","id":"5","color":"#fa7e19","latex":"K_{p}=\\left[K\\left(k_{a},k_{b}\\right)\\operatorname{for}k_{a}=\\left[0,0.01,...1\\right],k_{b}=\\left[0,0.25,...1\\right]\\right]"},{"type":"text","id":"14","text":"approximation of the polar"},{"type":"expression","id":"6","color":"#2d70b3","latex":"P_{0}=\\left[1\\left(k_{c},k_{d}\\right)\\operatorname{for}k_{c}=\\left[-1,-0.95,...1\\right]+0.1\\operatorname{random}\\left(\\right),k_{d}=\\left[-1,-0.95,...1\\right]+0.1\\operatorname{random}\\left(\\right)\\right]","pointOpacity":"P_{s}"},{"type":"expression","id":"7","color":"#c74440","latex":"D\\left(u,v\\right)=u.x\\cdot v.x+u.y\\cdot v.y"},{"type":"expression","id":"8","color":"#2d70b3","latex":"I_{P}\\left(u\\right)=\\prod_{k=1}^{\\operatorname{length}\\left(K_{p}\\right)}\\left\\{D\\left(u,K_{p}\\left[k\\right]\\right)\\le1:1,0\\right\\}"},{"type":"expression","id":"10","color":"#6042a6","latex":"P_{s}=\\left[I_{P}\\left(v\\right)\\operatorname{for}v=P_{0}\\right]"},{"type":"text","id":"16","text":"some established correspondences:\n- box centred around origin (edge length 2) <=> diamond around origin (edge length sqrt(2))\n- upper-right unit box (edge length 1) => lower-left region bounded by y = 1, x = 1, x + y = 1\n- disk centred at origin (radius r) <=> disk centred at origin (radius 1/r)\n- circle (unfilled) centred at origin (radius r) => disk centred at origin (radius 1/r)\n- box just above origin (edge length 2 horizontal, 1 vertical) => centre-bottom region bounded by x = -1, x = 1, y + x = 1, y - x = 1\n- vertical segment from (0, 0) to (0, 1) => bottom region bounded by y = 1\n- vertical segment from (0, -a) to (0, b) => centre region bounded by y = -1/a, y = 1/b\n- vertical segment from (a, -c) to (a, b) => region bounded by x - (c/a)*y = 1/a, x + (b/a)*y = 1/a\n- disk centred away from origin => region bounded by a conic section\n- n-sided regular polygon centred at origin, with edges r away from origin => n-sided polygon centred at origin, rotated by pi/n, with vertices 1/r away from origin"}]}}