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I want to compute the centroid of a body of the following form: $$ E\cap H_1\cap \cdots \cap H_T $$ where:

  • $E$ is an ellipsoid in $\mathbb{R}^d$: $E = \{x = c + Bu | u^T u \leq 1\}$, where $c$ is the ellipsoid center and $B$ is a matrix.
  • $H_1,\ldots,H_T$ are half-spaces.

Can this be done in time polynomial in $d$ (the dimension) and $T$ (the number of half-spaces)?

The motivation to this question comes from the ellipsoid method. In that method, after intersecting the current ellipsoid with a half-space, we find another ellipsoid that contains the resulting "half-ellipsoid". I would like to understand whether this last step is essential, or maybe we could just keep the half-ellipsoid and find its centroid, etc., without covering them with ellipsoids.

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