Usually, a regular plane curve X : I → ℝ² defined on an open interval is called convex if, for all t ∈ I, the trace X(I) lies entirely on one side of the closed half-plane determined by the tangent line at X(t) ([6]). A regular plane curve X : I → ℝ² is called locally convex if, for each t ∈ I there exists an open subinterval J ⊂ I containing t such that the curve X|_J restricted to J is a convex curve.

Hereafter, we will say that a locally convex curve X in the plane ℝ² is strictly locally convex if the curve is smooth (that is, of class C⁽³⁾) and is of positive curvature κ with respect to the unit normal N pointing to the convex side. Hence, in this case we have κ(s) = 〈X′(s), N(X(s))〉 > 0, where X(s) is an arc-length parametrization of X.

When f : I → ℝ is a smooth function defined on an open interval, we will also say that f is strictly convex if the graph of f has positive curvature κ with respect to the upward unit normal N. This condition is equivalent to the positivity of f″(x) on I.

Suppose that X denotes a strictly locally convex C⁽³⁾ curve in the plane ℝ² with the unit normal N pointing to the convex side. For a fixed point P = A ∈ X and a sufficiently small number h > 0, we consider the line m passing through P +hN(P) which is parallel to the tangent ℓ to X at P and the points A₁ and A₂ where the line m intersects the curve X.

Let us denote by ℓ₁, ℓ₂ the tangent lines of X at the points A₁,A₂ and by B,B₁,B₂ the points of intersection ℓ₁ ∩ ℓ₂, ℓ ∩ ℓ₁, ℓ ∩ ℓ₂, respectively. We let L_P (h), ℓ_P (h) and H_P (h) denote the lengths |A₁A₂| and |B₁B₂| of the corresponding segments and the height of the triangle △BA₁A₂ from the vertex B to the edge A₁A₂, respectively.

We also consider T_P(h),U_P(h), V_P(h) and W_P(h) defined by the area | △ AA₁A₂|, | △ BB₁B₂|, | △ BA₁A₂| of corresponding triangles and the area |□A₁A₂B₂B₁| of trapezoid □A₁A₂B₂B₁, respectively. Then, obviously we have

and

If we put S_P (h) the area of the region bounded by the curve X and chord A₁A₂, then we have ([18])

It is well known that parabolas satisfy the following properties

Proposition 1.1

Suppose that X denotes an open arc of a parabola. For an arbitrary point P ∈ X and a sufficiently small number h > 0, it satisfies

and

In fact, Archimedes showed that parabolas satisfy (1.1) ([25]).

Recently, in [18] the first and third authors of the present paper and others proved that (1.1) is a characteristic property of parabolas and established some characterizations of parabolas, which are the converses of well-known properties of parabolas originally due to Archimedes ([25]). For the higher dimensional analogues of some results in [18], see [16] and [17]. Some characterizations of hyperspheres, ellipsoids, elliptic hyperboloids, hypercylinders and W-curves in the (n+1)-dimensional Euclidean space ⁿ⁺¹ were given in [1, 4, 7, 8, 13, 15, 22]. In [19], some characteristic properties for hyperbolic spaces embedded in the Minkowski space were established.

In [12], it was shown that (1.2) is also a characteristic property of parabolas, which gives an affirmative answer to Question 3 in [21]. Thus, we have

Proposition 1.2

Suppose that X denotes a strictly locally convex C⁽³⁾curve in the plane ℝ². Then X is an open arc of a parabola if and only if it satisfies one of the following conditions.

• For all P ∈ X and sufficiently small h > 0, (1.4)SP(h)=λ(P)TP(h),

  where λ(P) is a function of P.

• For all P ∈ X and sufficiently small h > 0, (1.5)UP(h)=η(P)TP(h),

  where η(P) is a function of P.

Theorem 1.3

Let X denote a strictly locally convex C⁽³⁾curve in the plane ℝ². Then the following are equivalent.

• There exists a function ν(P) of P ∈ X such that for all P ∈ X and sufficiently small h > 0 the curve X satisfies(1.6)SP(h)=ν(P)UP(h).

• X is an open arc of a parabola.

In Theorem 1.3, obviously we have ν(P) = 8/3.

Combining (1.4) and (1.6), it follows from (1.1) and (1.3) that for an arbitrary point P and a sufficiently small number h > 0, parabolas satisfy

whenever λ(P) and ν(P) are functions of P with

We now naturally raise a question as follows:

Question 1.4

Suppose that X denotes a strictly locally convex C⁽³⁾curve in the plane ℝ²satisfying (1.7) for some functions λ(P) and ν(P) of P with (1.8). Then, is it an open arc of a parabola?

In Section 4, we give a partial affirmative answer to Question 1.4 as follows:

Theorem 1.5

Suppose that X denotes a strictly locally convex C⁽³⁾curve in the plane ℝ². Then the following are equivalent.

• There exist functions λ(P) and ν(P) of P ∈ X with(1.9)ν=ν(P)∈ℝ-(0,827]

  such that for all P ∈ X and sufficiently small h > 0 the curve X satisfiesSP(h)=λ(P)TP(h)+ν(P)UP(h).

• X is an open arc of a parabola.

In Theorem 1.5, λ(P) and ν(P) necessarily satisfy (1.8).

In [5], it was shown that parabolas satisfy T_P (h) = 2U_P (h) for all points P and h > 0. This property of parabolas was proved to be a characteristic one of parabolas ([12, 21, 23]). For some characterizations of parabolas or conic sections by properties of tangent lines, see [9] and [20]. In [14], using curvature function κ and support function h of a plane curve, the first and third authors of the present paper gave a characterization of ellipses and hyperbolas centered at the origin.

Among the graphs of functions, in [2, 3] Á. Bényi et al. gave some characterizations of parabolas. B. Richmond and T. Richmond established a dozen characterizations of parabolas using elementary techniques ([24]). In their papers, a parabola means the graph of a quadratic polynomial in one variable.

Throughout this article, all curves are of class C⁽³⁾ and connected, unless otherwise mentioned.

In order to prove Theorems 1.3 and 1,5 in Section 1, we need the following lemma.

Lemma 2.1

Suppose that X denotes a strictly locally convex C⁽³⁾curve in the plane ℝ²with the unit normal N pointing to the convex side. Then we have

and

where κ(P) is the curvature of X at P with respect to the unit normal N.

Lemma 2.2

Suppose that X denotes a strictly locally convex C⁽³⁾curve in the plane ℝ²with the unit normal N pointing to the convex side. Then we have

where LP′(h) means the derivative of L_P (h) with respect to h.

Lemma 2.3

Suppose that X denotes a strictly locally convex C⁽³⁾curve in the plane ℝ². Then, the height function H_P (h) satisfies

In this section, we prove Theorem 1.3.

It is trivial to show that any open arcs of parabolas satisfy 1) in Theorem 1.3 with ν(P) = 8/3.

Conversely, suppose that X denotes a strictly locally convex C⁽³⁾ curve in the plane ℝ² which satisfies for all P ∈ X and sufficiently small h > 0

where ν = ν(P) is a function of P ∈ X. Then, it follows from Lemma 2.1 that ν = 8/3.

We fix an arbitrary point P ∈ X.

Since 2U_P (h) = (H_P (h) − h)ℓ_P (h), (2.8) shows that

where the second equality follows from (2.9). Hence we see that (1.6) becomes

By differentiating (3.2) with respect to h, we get

On the other hand, by differentiating (2.9) with respect to h, we obtain

Hence, using (2.9) and (3.4), (3.3) may be rewritten as

Dividing the both sides of (3.3) by L_P (h), we have

Let us denote by y = f(x) the height function H_P (x) for sufficiently small x > 0. Then, it follows from (3.6) that the function y = f(x) satisfies

which is a homogeneous differential equation.

If we put v = y/x, from (3.7) we get

where we denote

It follows from Lemma 2.3 that

Note that

where

We consider two cases as follows.

Case 1

Suppose that dv/dx = 0 on an interval (0, ε) for some ε > 0.

Then v is constant and hence it follows from (3.10) that v = 2. This shows that

and hence using (2.9), (3.13) becomes 2hLP′(h)=LP(h). By integrating this equation we get

Integrating (3.14) implies that

Together with (3.14), (3.15) shows that at the point P for sufficiently small h > 0 the curve X satisfies

Case 2

Suppose that dv/dx ≠ 0.

In this case, the differential equation (3.8) becomes

which may be rewritten as

where we put

By integrating (3.17), we get

where C is a constant of integration. By letting x → 0, Lemma 2.3 yields that the left hand side of (3.18) tends to zero, which implies that the constant C must be zero. Hence v must be 2. This contradiction shows that this case cannot occur.

Combining Cases 1 and 2, we see that at the point P ∈ X for sufficiently small h > 0 the curve X satisfies

Since P ∈ X was arbitrary, Proposition 1.2 completes the proof of Theorem 1.3.

In this section, we prove Theorem 1.5.

It follows from Proposition 1.1 that any open arcs of parabolas satisfy 1) in Theorem 1.5.

Conversely, suppose that X is a strictly locally convex C⁽³⁾ curve in the plane ℝ² which satisfies for all P ∈ X and sufficiently small h > 0

where λ = λ(P) and ν = ν(P) are some functions of P ∈ X with

Then, it follows from Lemma 2.1 that for all P ∈ X the curve X satisfies

Now, we fix an arbitrary point P ∈ X.

From now on, we may assume that ν = ν(P) ≠ 0 because otherwise, at the point P the curve X satisfies (1.1) for all sufficiently small h > 0. Hence we assume that ν < 0 or ν > 8/27.

Since 2T_P (h) = hL_P (h) and U_P (h) is given by (3.1), from (1.7) we get

By differentiating (4.1) with respect to h, we obtain

Together with (2.9), (3.4) shows that (4.2) may be rewritten as

Multiplying the both sides of (4.3) by H_P (h)²/L_P (h), we have

Or equivalently, we get

Let us denote by y = f(x) the height function H_P (x) for sufficiently small x > 0. Then, it follows from (4.5) that the height function y = f(x) satisfies

where we put a = 2 − λ+ν = 3ν/2+2/3. By letting v = y/x, from (4.6) we obtain

where we denote

We decompose g(v) as follows:

where we put

with α < 0 and α < β. Since ν ≠ 8/27, the polynomial g(v) has distinct three roots 2, α and β.

We consider two cases as follows.

Case 1

Suppose that dv/dx = 0 on an interval (0, ε) for some ε > 0.

Then v is constant and hence it follows from Lemma 2.3 that v = 2. As in the proof of Case 1 in Section 3, we may prove that at the point P the curve X satisfies (1.1) for sufficiently small h > 0.

Case 2

Suppose that dv/dx ≠ 0.

Note that the polynomial g(v) has distinct three roots 2, α and β. The differential equation (4.7) becomes

Or equivalently, we get

where we put

and

Integrating (4.12) yields

where C is a constant of integration. It follows from ν < 0 or ν > 8/27 that η > 0. Hence, by letting x → 0, the left hand side of (4.15) goes to zero, which implies that C is zero and hence v = 2. This contradiction shows that this case cannot occur.

Combining Cases 1 and 2 shows that at the point P ∈ X for sufficiently small h > 0 the curve X satisfies

Since P ∈ X was arbitrary, Proposition 1.2 completes the proof of Theorem 1.5.