In the plane, for 0<q<2, equal dual curvature measures force origin-symmetric convex bodies to coincide; for q>n in R^n, distinct origin-symmetric bodies can share the same dual curvature measure.
A letter: The log-Brunn-Minkowski inequality for complex bodies
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In this short note we explain why the log-Brunn-Minkowski conjecture is correct for complex convex bodies. We do this by relating the conjecture to the notion of complex interpolation, and appealing to a general theorem by Cordero-Erausquin.
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The uniqueness and non-uniqueness of solutions to the even dual Minkowski problem
In the plane, for 0<q<2, equal dual curvature measures force origin-symmetric convex bodies to coincide; for q>n in R^n, distinct origin-symmetric bodies can share the same dual curvature measure.