Variation inequalities for smartingales

A result by N.G. Makarov [Algebra i Analiz, 1989] states that for martingales $(M_n)$ on the torus we have the strict inequality $$\liminf_{n\to\infty} \frac{M_n}{\sum_{k=1}^n |\Delta M_k|} > 0$$ on a set of Hausdorff dimension one, denoting by $\Delta M_n$ the martingale differences $\Delta M_n = M_n - M_{n-1}$. We discuss an extension of this inequality to so-called smartingales on convex, compact subsets of $\mathbb R^d$, which are piecewise polynomial (or spline) versions of martingales. As a tool we need and prove an estimate for smartingales in the spirit of the law of the iterated logarithm.