EXTREMUM PROBLEMS WITH BOUNDARY CONDITIONS. CONNECTION BETWEEN STEINER’S PROBLEM AND THE ISOPERIMETRIC PROBLEM - MAXIMA AND MINIMA - What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

CHAPTER VII. MAXIMA AND MINIMA

*§9. EXTREMUM PROBLEMS WITH BOUNDARY CONDITIONS. CONNECTION BETWEEN STEINER’S PROBLEM AND THE ISOPERIMETRIC PROBLEM

Interesting results arise in extremum problems when the domain of the variable is restricted by boundary conditions. The theorem of Weierstrass that in a compact domain a continuous function attains a largest and smallest value does not exclude the possibility that the extreme values are attained at the boundary of the domain. A simple, almost trivial, example is afforded by the function u = x. If x is not restricted and may range from – ∞ to + ∞, then the domain B of the independent variable is the entire number axis; and hence it is understandable that the function u = x has no largest or smallest value anywhere. But if the domain B is limited by boundaries, say 0 ≤ x ≤ 1, then there exists a largest value, 1, attained at the right endpoint, and a smallest value, 0, attained at the left endpoint. However, these extreme values are not represented by a summit or a depression in the curve of the function; they are not extrema relative to a full two-sided neighborhood. They change as soon as the interval is extended, because they remain at the endpoints. For a genuine peak or depression of a function, the extremal character always refers to a full neighborhood of the point where the value is attained; it is not affected by slight changes of the boundary. Such an extremum persists even under a free variation of the independent variable in the domain B, at least in a sufficiently small neighborhood. The distinction between such “free” extrema and those assumed at the boundary is illuminating in many apparently quite different contexts. For functions of one variable, of course, the distinction is simply that between monotone and non-monotone functions, and thus does not lead to particularly interesting observations. But there are many significant instances of extrema attained at the boundary of the domain of variability by functions of several variables.

This may occur, for example, in Schwarz’s triangle problem. There the domain of variability of the three independent variables consists of all triples of points, one on each of the three sides of the triangle ABC. The solution of the problem involved two alternatives: either the minimum is attained when all three of the independently variable points P, Q, R lie inside the respective sides of the triangle, in which case the minimum is given by the altitude triangle, or the minimum is attained for the boundary position when two of the points P, Q, R coincide with the common endpoint of their respective intervals, in which case the minimum inscribed “triangle” is the altitude from this vertex, counted twice. Thus the character of the solution is quite different according to which of the alternatives occurs.

In Steiner’s problem of the three villages the domain of variability of the point P is the whole plane, of which the three given points A, B, C may be considered as boundary points. Again there are two alternatives yielding two entirely different types of solutions: either the minimum isattained in the interior of the triangle ABC, which is the case of the three equal angles, or it is attained at a boundary point C. A similar pair of alternatives exists for the complementary problem.

As a last example we may consider the isoperimetric problem modified by restrictive boundary conditions. We shall thus obtain a surprising connection between the isoperimetric problem and Steiner’s problem and at the same time what is perhaps the simplest instance of a new type of extremum problem. In the original problem the independent variaable, the closed curve of given length, can be arbitrarily varied from the circular shape, and any such deformed curve is admissible into the competition, so that we have a genuine free minimum. Now let us consider the following modified problem: the curves C under consideration shall include in their interior, or pass through, three given points, P, Q, R, the area A is prescribed, and the length L is to be made a minimum. This represents a genuine boundary condition.

It is clear that, if A is prescribed sufficiently large, the three points P, Q, R will not affect the problem at all. Whenever the circle circumscribed about the triangle PQR has an area less than or equal to A, the solution will simply be a circle of area A including the three points. But what if Ais smaller? We state the answer here but omit the somewhat detailed proof, although it would not be beyond our reach. Let us characterize the solutions for a sequence of values of A which decreases to zero. As soon as A falls below the area of the circumscribed circle, the original isoperimetric circle breaks up into three arcs, all having the same radius, which form a convex circular triangle with P, Q, R as vertices (Fig. 232). This triangle is the solution; its dimensions can be determined from the given value of A. If A decreases further, the radius of these arcs will increase, and the arcs will become more and more nearly straight, until when A is exactly the area of the triangle PQR the solution is the triangle itself. If A now becomes even smaller, then the solution will again consist of three circular arcs having the same radius and forming a triangle with corners at P, Q, R. This time, however, the triangle is concave and the arcs are inside the triangle PQR (Fig. 233). As A continues to decrease, there will come a moment when, for a certain value of A, two of the concave arcs become tangent to each other in a corner R. With an additional decrease of A, it is no longer possible to construct a circular triangle of the previous type. A new phenomenon occurs: the solution is still given by a concave circular triangle, but one of its corners R′ has become detached from the corresponding corner R, and the solution now consists of a circular triangle PQR′ plus the straight line RR′ counted twice (because it travels from R′ to R and back). This straight segment is tangent to the two arcs tangent to each other at R′. If A decreases further, the separation process will also set in at the other vertices. Eventually we obtain as solution a circular triangle consisting of three arcs of equal radius tangent to each other and forming an equilateral circular triangle P′Q′R′, and in addition three doubly counted straight segments P′P, Q′Q, R′R (Fig. 234). If, finally, A shrinks to zero, then the circular triangle reduces to a point, and we return to the solution of Steiner’s problem; the latter is thus seen to be a limiting case of the modified isoperimetric problem.

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Fig. 231

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Fig. 232

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Fig. 233

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Fig. 234

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Fig. 235

Figs. 231–5 Isoperimetrio figures tending to the solution of Steiner’s problem.

If P, Q, R form an obtuse triangle with an angle of more than 120°, then the shrinking process leads to the corresponding solution of Steiner’s problem, for then the circular arcs shrink toward the obtuse vertex. The solutions of the generalized Steiner problem (see Figs. 216–8 on p. 360) may be obtained by limiting processes of a similar nature.