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The following problem is listed in Steven Lay's "Convex Sets and Their Applications" (1982) as unsolved (paraphrased):

Let $B$ be the unit ball in $\mathbb{R}^3$ and $0 < V < \pi$. Define $\mathcal{F}$ as the family of all convex subsets of $B$ with volume $V$. Find the member of $\mathcal{F}$ with maximum surface area.

The conjectured answer (as of 1982) is $B \cap$ { $(x_1,x_2,x_3)\in \mathbb{R}^3 | |x_1| < c $} for appropriately chosen $c$.

Does anyone know if this problem has been solved, or if any progress has been made on it? I couldn't find any recent references in the literature to it.

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Is the corresponding 2D problem solved? – Gjergji Zaimi Mar 25 '11 at 4:54
    
I'm sure the solution is known for the 2D problem, but I can't find a source for the solution at the moment. I'm fairly sure the maximum achieved by the 2D analogue of the conjectured set, but not totally sure. – Logan Maingi Mar 27 '11 at 19:40

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