Geometric Interpretation
In two dimensions, 2x1 + 2x2 is the perimeter of a rectangle with sides of length x1 and x2. Similarly, 4√x1x2 is the perimeter of a square with the same area. Thus for n = 2 the AM–GM inequality states that only the square has the smallest perimeter amongst all rectangles of equal area.
The full inequality is an extension of this idea to n dimensions. Every vertex of an n-dimensional box is connected to n edges. If these edges' lengths are x1, x2, . . ., xn, then x1 + x2 + · · · + xn is the total length of edges incident to the vertex. There are 2n vertices, so we multiply this by 2n; since each edge, however, meets two vertices, every edge is counted twice. Therefore we divide by 2 and conclude that there are 2n−1n edges. There are equally many edges of each length and n lengths; hence there are 2n−1 edges of each length and the total edge-length is 2n−1(x1 + x2 + · · · + xn). On the other hand,
is the total length of edges connected to a vertex on an n-dimensional cube of equal volume. Since the inequality says
we get
with equality if and only if x1 = x2 = · · · = xn.
Thus the AM–GM inequality states that only the n-cube has the smallest sum of lengths of edges connected to each vertex amongst all n-dimensional boxes with the same volume.
Read more about this topic: Inequality Of Arithmetic And Geometric Means
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