We establish new functional versions of the Blaschke-Santaló inequality on the volume product of a convex body which generalize to the nonsymmetric setting an inequality of K. Ball [2] and we give a simple proof of the case of equality. As a corollary, we get some inequalities for logconcave functions and Legendre transforms which extend the recent result of Artstein, Klartag and Milman [1], wi...

The singular points of an irreducible plane cubic curve are quite limited: one knot/node, or one cusp. Our research starts originally with the Descartes Folium, which has a knot/node, and is able to have many group structures. The original results are concentrated in six directions: (i) special structures on affine algebraic varieties, (ii) theory of K-groups, (iii) isomorphisms of K-groups, (i...