An α-concave measure with prescribed Euclidean surface area measure exists when the target measure has finite first moment, zero barycenter, and full-dimensional support; the true α-concave function version remains open.
H., Villa, R.:Rogers–Shephard inequality for log- concave functions
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A Minkowski problem for $\alpha$-concave functions via optimal transport
An α-concave measure with prescribed Euclidean surface area measure exists when the target measure has finite first moment, zero barycenter, and full-dimensional support; the true α-concave function version remains open.