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Minkowski's Theorem asserts that every centered measure on the sphere whichis not concentrated on a great subsphere is the surface area measure of someconvex body, and, moreover, the surface area measure determines a convex bodyuniquely.  In this manuscript we prove an extension of Minkowski's theorem. Consider ameasure $\mu$ on $\mathbb{R}^n$ with positive degree of concavity and positivedegree of homogeneity. We show that a surface area measure of a convex set $K$,weighted with respect to $\mu$, determines a convex body uniquely up to$\mu$-measure zero. We also establish an existence result under naturalconditions including symmetry.  We apply this result to extend the solution to classical Shephard's problem,which asks the following: if one convex body in $\mathbb{R}^n$ has largerprojections than another convex body in every direction, does it mean that thevolume of the first convex body is also greater? The answer to this question isaffirmative when $n\leq 2$ and negative when $n\geq 3.$  In this paper we introduce a new notion which relates projections of convexbodies to a given measure $\mu$, and is a direct generalization of the Lebesguearea of a projection. Using this notion we state a generalization of theShephard problem to measures and prove that the answer is affirmative for$n\leq 2$ and negative for $n\geq 3$ for measures which have a positive degreeof homogeneity and a positive degree of concavity. We also prove stability andseparation results, and establish useful corollaries. Finally, we describe twotypes of uniqueness results which follow from the extension of Minkowski'stheorem.

An extension of Minkowski's theorem and its applications to questions about projections for measures