Shifting lsoperimetric problem: With fixed area, what plane figure has the smallest perimeter? Steiner symmetrization
With fixed area, what plane figure has the smallest perimeter? Shifting Steiner symmetrization Isoperimetric problem:
Erdos-Ko-Rado Theorem Let FC(),n≥2k. S,T∈F,SnT≠0->lF≤ induction on n and k F={S∈F|n∈S] Fo={S∈F|n年S} F1={S\{n}|S∈F} Fo()1 . (-) intersecting F() intersecting? F引≤(2) IFI=Fol+=Fol+1Fls()+()=(
Let F [n] k ⇥ , n ⇥ 2k. |F| ⇥ n 1 k 1 ⇥ Erdős-Ko-Rado Theorem ⇤S, T F, S ⌃ T ⇥= ⌅ induction on n and k F0 = {S F | n ⇥ S} F1 = {S F | n S} |F| = |F0| + |F1| F 1 = {S \ {n} | S F1} = |F0| + |F 1| F0 [n1] k ⇥ F⇥ 1 [n1] k1 ⇥ intersecting I.H. |F0| n2 k1 ⇥ |F⇥ 1| n2 k2 ⇥ intersecting? I.H. n2 k1 ⇥ + n2 k2 ⇥ = n1 k1 ⇥
Shifting (compression) F remains intersecting after deleting n
Shifting (compression) special F [n] k ⇥ F remains intersecting after deleting n n i