The mincing problem

How many cuts it take to mince a clove of garlic into one million pieces?

A professional chef notes that the usual method will do: First, she slices the garlic one-thousand times along its length and one-thousand times across its width.  This gives

\[ 1000\times 1000 = 1\,000\,000\]

tiny little garlic pieces with only two-thousand cuts. (Our characters remain intentionally two-dimensional—a three-dimensional chef would rotate the garlic to slice along its height, mincing the garlic into one-million pieces with only three-hundred cuts of her knife.)

A naïve chef claims a better way. “First,” he says, “I slice the garlic in two. Then, I cut each of those pieces in half with a single slice.” He continues along, carefully rearranging the garlic so that each run of his blade divides every piece in two again.

“See?” he points. “Each of my slices doubles the number of garlic bits.”

“Two! Four! Eight!” he shouts. “With only twenty slices, I’ll mince it to a million minuscule morsels.”

Then there’s us: sloppy, lazy cooks. Unlike the professional chef, we can’t reliably cut a clove of garlic into one-hundred thin slices without risking our fingers. And, we note wryly, it’s only a matter of time before the cocksure chef discovers a problem with his approach: the final cut requires half a million fragments of garlic arranged into a perfect line!

Instead of risking our fingers or sanity, we’re going to use a technique known to professionals and amateurs alike: we will chop the garlic at random, over and over, until we generate one million pieces. How many chops does our simple approach require?

The answer: Our rustic chop will mince almost as quickly as the professional’s honed technique. Put mathematically, the number of cells in our random chop also grows quadratically with the number of slices we make. With \(n\) slices, our technique produces \(n + n(n-1)/ 4\pi\) bits of garlic when the garlic is a circle. Taking \(n=3540\) yields, on average, just over one million pieces of garlic. Neat!

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Strictly convex surfaces in 3D have an equicurved point

The two principal radii of curvature at a point on a surface in three dimensions provide a succinct description of the local geometry of the surface. The principal curvatures determine whether a surface is locally flat, convex, or a saddle (Figure 1). The radii (plural of radius, pronounced ray-dee-eye or ˈrādēˌī) prove even more dispositive in the theory of convex surfaces. Convex surfaces necessarily have positive radii of curvature everywhere, but much more is true: Knowing that the two radii of curvature at every point on a convex surface in fact determines the entire surface up to translations. But when does the reverse hold?  Continue reading

Convexity and positive functions

In this post, we offer a proof of another classical result: A sufficiently smooth \(1\)-homogeneous function \(h\colon \mathbb{R}^2\to \mathbb{R}\) is convex if and only if

\[h(\theta) + h”(\theta) \ge 0 \text{ for all } 0 \le \theta < 2\pi,\]

where \(h(\theta):= h(\cos(\theta), \sin(\theta)\) represents evaluation of \(h\) on the unit circle. Since one-homogeneous convex functions are support functions, the equation above characterizes convex bodies in two dimensions.

The non-standard proof below provides a more modern route to the result than classical approaches. I expect that this technique extends to higher dimensions more directly than the classical arguments available, e.g., in Groemer’s text.

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Meissner bodies: A new proof of an old result

Bodies of constant width fascinate mathematicians and artists alike. These elegant shapes roll between two parallel surfaces like ball bearings, yet they can be very far from spherical. These counterintuitive objects, variously called spheroforms, orbiforms, or Gleichdick, depending on who you ask, remain mysterious, despite the intense interest of mathematicians over the last century.
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