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Given

$$X^{(n)}_{s,t} = \int_{s < r_1 < \cdots < r_n < t}\dot{X}_{r_1} \otimes \cdots \otimes \dot{X}_{r_n}dr_1\cdots dr_n$$,

I need to show the following bound

$$ |X^{(n)}_{s,t}| \leq \int_{s < r_1 < \cdots < r_n < t} |\dot{X}_{r_1}|\cdots|\dot{X}_{r_n}|dr_1\cdots dr_n \leq \frac{||\dot{X}||_{\infty}^n}{n!}|t-s|^n $$

where $X:[0,T] \rightarrow V$ is a smooth path.

I think that this requires proof by induction, but I'm not able to do it. Would someone mind helping? Thanks

Jamal
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  • I'm not sure exactly what the objects involved in the question are. But it seems that the first inequality follows from the triangle inequality for integrals ("pulling in" the absolute values to the integral), and the second is bounding an integral of a function f over a region A by the maximum of f times the volume of region A. The right hand side would then follow from computing the volume of an $n$-simplex; see e.g. https://math.stackexchange.com/questions/1718021/intuition-for-volume-of-a-simplex-being-frac-1n. – Chris Mar 06 '23 at 20:26
  • @Chris thank you very much, the $n!$ was the bit I couldn't wrap my head around so that link is very much appreciated! – Jamal Mar 06 '23 at 22:03

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