Let $B_{p_0,p_1,...,p_n}$ be a $n$-th degree concave Bézier curve on $[0,1]$ determined by control points $p_0,p_1,...,p_n$. Let the control points be all non-negative. Such curve admits an explicit definition in terms of Bernstein basis polynomials $b_{j,n}(t): = {n \choose j} t^j (1-t)^{n-j}$, $j \in \{ 0, 1, ..., n\}$, i.e.

$ B_{p_0,p_1,...,p_n}(t)=\sum_{j=0}^n p_j b_{j,n}(t), \quad t \in [0,1]. $

My question is maybe a trivial one: is there a way to obtain "accurate" upper and lower bounds for the distance between the curve and the $j$-th control point at $t=j/n$, i.e. $p_j -B_{p_0,p_1,...,p_n}(j/n)$, $j=0,1,...,n$? I think the simple ones that I have found are too crude:

$ p_j - B_{p_0,p_1,...,p_n}(j/n) \geq p_j( 1 - b_{j,n}(j/n)) - nb_{j,n}(j/n) \max_{0\leq i \leq n} p_i $

$ p_j - B_{p_0,p_1,...,p_n}(j/n) \leq p_j( 1 - b_{j,n}(j/n)) - n \left[b_{0,n}(j/n)\mathbb{1}_{\{j\geq n/2\}}+ b_{n,n}(j/n)\mathbb{1}_{\{j < n/2\}} \right]\min_{0\leq i \leq n} p_i $

and better ones can be found. Am I missing something?


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