Abstract
For a convex body K ⊂ ℝn and i ∈ {1, …, n − 1}, the function assigning to any i-dimensional subspace L of ℝn, the i-dimensional volume of the orthogonal projection of K to L, is called the i-th projection function of K. Let K, K 0 ⊂ ℝn be smooth convex bodies with boundaries of class C 2 and positive Gauss-Kronecker curvature and assume K 0 is centrally symmetric. Excluding two exceptional cases, (i, j) = (1, n − 1) and (i, j) = (n − 2, n − 1), we prove that K and K 0 are homothetic if their i-th and j-th projection functions are proportional. When K 0 is a Euclidean ball this shows that a convex body with C 2 boundary and positive Gauss-Kronecker with constant i-th and j-th projection functions is a Euclidean ball.
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The second author was supported in part by the European Network PHD, FP6 Marie Curie Actions, RTN, Contract MCRN-511953.
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Howard, R., Hug, D. Smooth convex bodies with proportional projection functions. Isr. J. Math. 159, 317–341 (2007). https://doi.org/10.1007/s11856-007-0049-z
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DOI: https://doi.org/10.1007/s11856-007-0049-z