# Convex bodies with symmetric shadows.

Theorem. If all orthogonal projections of a convex body $K \subset \mathbb{R}^n$ onto $2$-dimensional subspaces have a center of symmetry, then $K$ has a center of symmetry.

This is a classic result of Blaschke and Hessenberg (that I just learned thanks to Guillaume's comment.). A short simple proof of it can be found in Bonnesen and Fenchel.

I wonder if it is necessary to know what happens for every orthogonal projection or whether we can get by with less:

Question 1. Let $K \subset \mathbb{C}^{n}$ be a convex body. Assume all orthogonal projections of $K$ onto complex lines have a center of symmetry. Does it follow that $K$ must also have a center of symmetry?

Note. The center of symmetry of the shadows may depend on the subspace containing it.

A similar question is:

Question 2. Let $K \subset \mathbb{C}^{n}$ be a convex body. Assume all orthogonal projections of $K$ onto Lagrangian subspaces have a center of symmetry. Does it follow that $K$ must also have a center of symmetry?

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For the first question, yes. This is a theorem due to Blaschke and Hessenberg, see Theorem 2 in C.A. Rogers, "Sections and projections of convex bodies", purl.pt/2464/1/j-5293-b-vol24-fasc2-art3_PDF/… – Guillaume Aubrun Jan 22 '13 at 14:52
You're right. Thanks!! The paper you mention seems to use the fact that all projections have the same center, but this is not necessary. I am editing my question to reflect your input. – alvarezpaiva Jan 22 '13 at 15:20