In this paper, we introduce the concept of a logarithmic convex structure. Let X be a set and D : X × X → [1, ∞) a function satisfying the following conditions: (i) For all x, y ∈ X, D(x, y) ≥ 1 and D(x, y) = 1 if and only if x = y. (ii) For all x, y ∈ X, D(x, y) = D(y, x). (iii) For all x, y, z ∈ X, D(x, y) ≤ D(x, z)D(z, y). (iv) For all x, y, z ∈ X, z ≠ x, y and λ ∈ (0, 1), D(z, W(x, y, λ)) ≤ D λ (x, z)D 1−λ (y, z), D(x, y) = D(x, W(x, y, λ))D(y, W(x, y, λ)), where W : X ×X ×[0, 1] → X is a continuous mapping. We name this the logarithmic convex structure. In this work we prove some fixed point theorems in the logarithmic convex structure.