Contributions to affine surface area

Representations of equiaffine surface area, due to Leichtweiß resp. Schütt &; Werner, are generalized top-affine surface area measures. We provide a direct proof which shows that these representations coincide. In addition, we establish two theoremes which in particular characterize all those convex bodies geometrically for which the affine surface area is positive. The present approach also leads to proofs of the equiaffine isoperimetric inequality and the Blaschke-Santaló inequality, including the characterization of the case of equality.     

The generalized L p -mixed affine surface area

In this article, we put forward the concept of the (i, j)-type L^p-mixed affine surface area, such that the notion of L^p-affine surface area which be shown by Lutwak is its special cases. Furthermore, applying this concept, the Minkowski inequality for the (i,-p)-type L^p-mixed affine surface area and the extensions of the well-known L^p-Petty affine projection inequality are established, respectively. Besides, we give an affirmative answer for the generalized L^p-Winterniz monotonicity problem.     

General affine surface areas

Two families of general affine surface areas are introduced. Basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for L? affine surface areas are established.     

Extremal general affine surface areas

For a convex body K in ${\mathbb{R}}^n$, we introduce and study the extremal general affine surface areas, defined by${\mathrm{\text{IS}}}_{\phi }(K):=\underset{K^{\prime }?K}{\sup }?{\mathrm{as}}_{\phi }(K),{\mathrm{\text{os}}}_{\psi }(K):=\underset{K^{\prime }?K}{\inf }?{\mathrm{as}}_{\psi }(K)$ where ${\mathrm{as}}_{\phi }(K)$ and ${\mathrm{as}}_{\psi }(K)$ are the $L_{\phi }$ and $L_{\psi }$ affine surface area of K, respectively. We prove that there exist extremal convex bodies that achieve the supremum and infimum, and that the functionals ${\mathrm{\text{IS}}}_{\phi }$ and ${\mathrm{\text{os}}}_{\psi }$ are continuous. In our main results, we prove Blaschke-Santaló type inequalities and inverse Santaló type inequalities for the extremal general affine surface areas. This article may be regarded as an Orlicz extension of the recent work of Giladi, Huang, Schütt and Werner (2020), who introduced and studied the extremal $L_p$ affine surface areas.     

Enclosed convex hypersurfaces with maximal affine area

In this paper we study the regularity of closed, convex surfaces which achieve maximal affine area among all the closed, convex surfaces enclosed in a given domain in the Euclidean 3-space. We prove the C^1,α regularity for general domains and C^1,1 regularity if the domain is uniformly convex. This work is supported by the Australian Research Council. Research of Sheng was also supported by ZNSFC No. 102033. On leave from Zhejiang University.     

An affine characterization of the Veronese surface

IfM ² is a nondegenerate surface in a 4-dimensional Riemannian manifold , then there is a natural affine metricg defined onM ². It is shown that this affine metricg is conformal to the induced Riemannian metric onM ² if and only ifM ² is a minimal submanifold of in the usual Riemannian sense. If the conformal factor is a constant, then the two metrics are said to be homothetic. It is shown that there does not exist a nondegenerate surface in Euclidean space ⁴ or hyperbolic spaceH ⁴ whose affine metric is homothetic to the induced Riemannian metric. Furthermore, ifM ² is a nondegenerate surface in the standard 4-sphereS ⁴ whose affine metric is homothetic to the induced Riemannian metric, thenM ² is a Veronese surface.T. Cecil was supported by NSF Grant No. DMS-9101961.     

The mixed Lp affine surface areas for functions

In this paper, we propose a definition of a general mixed L_p Affine surface area, ?n ≠ p ∈ ?, for multiple functions. Our definition is di?erent from and is “dual” to the one in [11] by Caglar and Ye. In particular, our definition makes it possible to establish an integral formula for the general mixed L_p Affine surface area of multiple functions (see Theorem 3.1 for more precise statements). Properties of the newly introduced functional are proved such as affine invariance, and related affine isoperimetric inequalities are proved.     

Geominimal surface area

Geometriae Dedicata -     

Mixed integer reformulations of integer programs and the affine TU-dimension of a matrix

Mathematical Programming - We study the reformulation of integer linear programs by means of a mixed integer linear program with fewer integer variables. Such reformulations can be solved...     

On random surface area

Consider a random smooth Gaussian field G(x): F ? \mathbbR F \to \mathbb{R} , where F is a compact set in \mathbbR^d {\mathbb{R}^d} . We derive a formula for the average area of a surface determined by the equation G(x) = 0 and give some applications. As an auxiliary result, we obtain an integral expression for the area of a surface determined by zeros of a nonrandom smooth field. Bibliography: 13 titles.     

Gibbs' phenomenon and surface area

If a function is of bounded variation on and is a positive approximate identity, we prove that the area of the graph of converges from below to the relaxed area of the graph of . Moreover we give asymptotic estimates for the area of the graph of the square partial sums of multiple Fourier series of functions with suitable discontinuities.

     

Mixed convection similarity solutions on a horizontal surface

The mixed convection boundary layer on a horizontal plate is considered for the two separate cases when there is a uniform free stream with the plate held fixed and when there is no outer flow but the plate is moving continuously with a uniform velocity along its length. In both cases we assume that power law temperature distribution on the plate which enables the governing equations to be reduced to similarity form. For the first problem we consider the range of buoyancy parameter for which there are dual solutions, showing how these dual solutions arise from a bifurcation and how the lower branch of solutions terminate as the buoyancy parameter tends to zero. For the second problem we show that there is a unique solution for all positive values of the buoyancy parameter and that for negative values the solution terminates at a singular solution with algebraic decay.

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Zusammenfassung Die Grenzschicht an einer horizontalen Platte mit gemischter Konvektion wird für zwei Fälle untersucht: Für eine ruhende Platte bei gleichförmiger Außenströmung und für eine Platte, die sich kontinuierlich mit konstanter Geschwindigkeit in ihrer eigenen Ebene bewegt, ohne Außenströmung. Für beide Fälle wählen wir ein Potenzgesetz für die Temperaturverteilung entlang der Wand, für welche die Grundgleichungen eine Ähnlichkeitslösung zulassen. In der ersten Problemstellung betrachten wir einen Bereich des Auftriebsparameters, für den zweifache Lösungen existieren. Wir zeigen, wie diese doppelten Lösungen aus einer Bifurkation entstehen und wie der untere Lösungszweig endet, wenn der Auftriebsparameter gegen Null strebt. Für die zweite Problemstellung zeigen wir, daß eine eindeutige Lösung existiert für alle positiven Werte des Auftriebsparameters und daß für negative Werte eine singuläre Lösung mit algebraischem Zerfall erreicht wird.

     

Algebraic monoids with affine unit group are affine

In this short paper we prove that any irreducible algebraic monoid whose unit group is an affine algebraic group is affine.     

Lorentzian affine hyperspheres with constant affine sectional curvature

We study affine hyperspheres with constant sectional curvature (with respect to the affine metric ). A conjecture by M. Magid and P. Ryan states that every such affine hypersphere with nonzero Pick invariant is affinely equivalent to either

or

where the dimension satisfies or . Up to now, this conjecture was proved if is positive definite or if is a -dimensional Lorentz space. In this paper, we give an affirmative answer to this conjecture for arbitrary dimensional Lorentzian affine hyperspheres.

     

Preserving affine baire classes by perfect affine maps

Let ? : X → Y be an affine continuous surjection between compact convex sets. Suppose that the canonical copy of the space of real-valued affine continuous functions on Y in the space of real-valued affine continuous functions on X is complemented. We show that if F is a topological vector space, then f : Y → F is of affine Baire class α whenever the composition f ? ? is of affine Baire class α. This abstract result is applied to extend known results on affine Baire classes of strongly affine Baire mappings.