Convexification and Potato Peeling

 You may have wondered, as I have, about the convexifying effect of peeling a potato, so it comes as a great relief that this is a well studied subject.  The opening paragraph of Goodman (1981) is priceless:

A swivel-type potato peeler, when used in the usual (and somewhat wasteful) way, produces a convex peeled potato, which is a subset of the original unpeeled and non-convex potato. This suggests, in an obvious way, the problem of maximizing the volume of a convex body contained in a given non-convex body. That such a largest convex subset exists follows at once from the Blaschke selection theorem (see Proposition 1 below). The problems then become: (1) What proportion of the original volume are we assured of saving? and (2) How do we determine the largest convex 'peeled potato' inside a given non-convex one ?

A swivel-type potato peeler, when used in the usual (and somewhat wasteful) way, produces a convex peeled potato, which is a subset of the original unpeeled and non-convex potato. This suggests, in an obvious way, the problem of maximizing the volume of a convex body contained in a given non-convex body. That such a largest convex subset exists follows at once from the Blaschke selection theorem (see Proposition 1 below). The problems then become: (1) What proportion of the original volume are we assured of saving? and (2) How do we determine the largest convex 'peeled potato' inside a given non-convex one ?

A swivel-type potato peeler, when used in the usual (and somewhat wasteful) way, produces a convex peeled potato, which is a subset of the original unpeeled and non-convex potato. This suggests, in an obvious way, the problem of maximizing the volume of a convex body contained in a given non-convex body. That such a largest convex subset exists follows at once from the Blaschke selection theorem (see Proposition 1 below). The problems then become: (1) What proportion of the original volume are we assured of saving? and (2) How do we determine the largest convex 'peeled potato' inside a given non-convex one ?A swivel-type potato peeler, when used in the usual (and somewhat wasteful) way, produces a convex peeled potato, which is a subset of the original unpeeled and non-convex potato. This suggests, in an obvious way, the problem of maximizing the volume of a convex body contained in a given non-convex body. That such a largest convex subset exists follows at once from the Blaschke selection theorem (see Proposition 1 below). The problems then become: (1) What proportion of the original volume are we assured of saving? and (2) How do we determine the largest convex 'peeled potato' inside a given non-convex one ?A swivel-type potato peeler, when used in the usual (and somewhat wasteful) way, produces a convex peeled potato, which is a subset of the original unpeeled and non-convex potato. This suggests, in an obvious way, the problem of maximizing the volume of a convex body contained in a given non-convex body. That such a largest convex subset exists follows at once from the Blaschke selection theorem (see Proposition 1 below). The problems then become: (1) What proportion of the original volume are we assured of saving? and (2) How do we determine the largest convex 'peeled potato' inside a given non-convex one ?