In this talk, we will speak on our recent work on the (dual) affine surface area. Specifically, by establishing the variational formula for the (dual) affine surface area, we define the affine (intersection or) projection measure. Taking into account the two important characters in convex geometry: the (intersection or) projection function and the ellipsoid, we establish new affine isoperimetric inequalities, which open up a new passage to attack the longstanding Lutwak conjecture on the extremum problem of affine quermassintegrals since 1980s. 

 

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