Worst Approximation

Abstract

We recall that the deviation (or excess) of a set G (assumed nonempty in this chapter, without any special mention) from an element XQ in a normed linear space X is the number δ(G, χ₀) ≥ 0 defined by

$$ \delta (G,x_0 ): = \mathop {sup}\limits_{g \in G} \parallel g - x_0 \parallel , $$

(2.1)

and any g₀∈G for which this sup is attained, i.e., such that

$$ \parallel g_0 - x_0 \parallel = \mathop {sup}\limits_{g \in G} \parallel g - x_0 \parallel , $$

(2.2)

or equivalently, such that

$$ \parallel g_0 - x_0 \parallel \geqslant \parallel g - x_0 \parallel (g \in G), $$

(2.3)

is called an element of worst approximation of (or a farthest point to) x₀in G (see Figure 2.1).