  Motivated by the Dikin walk, we develop aspects of an interior-point theory
for sampling in high dimension. Specifically, we introduce a symmetric
parameter and the notion of strong self-concordance. These properties imply
that the corresponding Dikin walk mixes in $\tilde{O}(n\bar{\nu})$ steps from a
warm start in a convex body in $\mathbb{R}^{n}$ using a strongly
self-concordant barrier with symmetric self-concordance parameter $\bar{\nu}$.
For many natural barriers, $\bar{\nu}$ is roughly bounded by $\nu$, the
standard self-concordance parameter. We show that this property and strong
self-concordance hold for the Lee-Sidford barrier. As a consequence, we obtain
the first walk to mix in $\tilde{O}(n^{2})$ steps for an arbitrary polytope in
$\mathbb{R}^{n}$. Strong self-concordance for other barriers leads to an
interesting (and unexpected) connection -- for the universal and entropic
barriers, it is implied by the KLS conjecture.

Data Structures and Algorithms