Lecture 17: Convex Bodies and Ellipsoid

Yesterday we talked about the center of gravity; since some of you had not seen that before, the formula can seem somewhat mysterious. But it’s the natural extension of the discrete case, which you may have seen before, e.g., even in Melanie’s talk at this Wednesday’s theory lunch. Suppose we have {N} objects in {{\mathbb R}^n}, the {i^{th}} one having location {x_i \in {\mathbb R}^n} and mass {\mu_i \geq 0}. Then the center of gravity (or the center of mass, or centroid) is defined as

\displaystyle  c = \frac{\sum_{i \in [N]} x_i \mu_i }{\sum_{i \in [N]} \mu_i}.

The continuous analog of this where we have a general measure {\mu(x)} over {{\mathbb R}^n} (basically replacing sums by integrals), is

\displaystyle  c = \frac{\int_{x \in R^n} x \; d\mu(x) }{\int_{x \in R^n} d\mu(x)}.

The numerator is the total measure over {{\mathbb R}^n}. (In class I was implcitly assuming the uniform measure over {K \subseteq {\mathbb R}^n}, which is given by {d\mu(x) = \mathbf{1}_{x \in K} dx}.

John’s questions: the {1/e} in Grunbaum’s theorem (that each hyperplane through the centroid of a convex body contains at least {1/e} fraction of the mass on either side) is indeed best possible for convex bodies. And the proof is clever but not difficult See Grunbaum’s (very short) paper for examples and proof, or these notes by Jon Kelner or Santosh Vempala.

Guru’s question: does the theorem hold for other measures, and not just for the uniform measure over a convex bodies? In retrospect, it’s easy to see I was dead wrong (and I even knew the answer, had I thought about it for a minute more, sorry): it does not. E.g., consider {n+1} equal point masses at the vertices of an {n}-dimensional simplex. No matter which point you choose, you can find a hyperplane through it that contains only a single point (which is {1/n+1} of the mass) on one side. Grunbaum actually shows (in the same paper) that you can find a point that ensures at least {1/(n+1)} fraction of the mass on either side. I wonder what minimal assumptions can we make on the measure to get back the {1/e}? Replies in the comments, please.

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