The Nakajima problem asks if a convex body with constant width and constant brightness must be a Euclidean ball. An affirmative answer is only known in three dimensions(Howard, 2006). A convex body has C^1 boundary if its boundary is locally the graph of a differentiable function. Suppose any C^1 convex body with constant width and constant brightness must be a Euclidean ball. From this, can it be shown that any convex body with constant width and constant brightness must be a ball?
Dear Geoff Diestel,
You can find partial answers to your question in the following articles:
(1) https://arxiv.org/pdf/math/0306437
See the corollary next to theorem 2.
(2)https://pdfs.semanticscholar.org/96c8/a472922c5a313203faa213d22a34e5d470b1.pdf
see corollary 1.2.
Best regards
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