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Let $K \subset \mathbf{R}^3$ be a compact set.

What is the smallest set $C$ containing $K$, with the property that in a neighbourhood of $C$, the closest-point projection of surfaces onto $C$ decreases area?

  • I think that it exists: a proof via Zorn's lemma seems conceivable.
  • If $K$ is convex, then $C = K$. In general, $C$ would be a subset of the convex hull of $K$.
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  • $\begingroup$ Do you want to say "the closest-point projection", or "there is a projection"? $\endgroup$
    – Anton Petrunin
    Commented Jan 10, 2023 at 10:27
  • $\begingroup$ @AntonPetrunin I meant the closest-point projection. Thanks for pointing this out, I've now fixed it. $\endgroup$
    – Leo Moos
    Commented Jan 10, 2023 at 10:30

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I doubt that it exists; let me explain why.

Consider a surface $\Sigma$ of lest area that surrounds $K$. It $\Sigma$ might contain a piece $\Omega$ of generic minimal surface. Note that that projection from equidistant piece to $\Omega$ is not area-decreasing so the body bounded by $\Sigma$ does not meet your condition.

On the other hand, it seems possible to approximate $\Sigma$ from outside by strictly mean-curvature-convex surfaces $\Sigma_n$. So the bodies bounded by $\Sigma_n$ satisfy the condition.

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  • $\begingroup$ I'm not quite convinced... Why is the projection onto an area-minimizing surface not area-decreasing? Also, I don't think that $\Sigma_n$ being strictly mean-convex is sufficient to ensure the area-decreasing property. (Picture for example a strictly mean-convex body around two horizontal disks lying a distance $\delta > 0$ apart - the area-minimizing catenoid sees its area increase when projected onto this.) $\endgroup$
    – Leo Moos
    Commented Jan 10, 2023 at 12:50
  • $\begingroup$ First part is done by computations --- if principle curvatures do not vanish then projection increases the area of equidistant surface. But, if the surface is strictly mean-curvature-convex, then the projection of a small nbhd is area-decreasing --- again, by a straightforward computations. $\endgroup$
    – Anton Petrunin
    Commented Jan 10, 2023 at 16:03

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